Unit 1
Chapter 1
Data Analysis
1
AP Statistics Handout: Lesson 1.1
Topics: quantitative and categorical data, misleading graphs
Lesson 1.1 Guided Notes
Quantitative vs. Categorical Data
Quantitative data: Data that is _________________ (think ‘quantities’). Usually a __________________.
You can line the values up in order.
Ex: weight, # of AP classes, SAT score, blood pressure, income, yards per catch, etc.
Categorical data: Data that fits an individual into one of several categories that don’t _______________.
Usually represented by counts or percentages, and isn’t a number (but it can be).
Ex: eye color, race, gender, social security number, and zip code.
Graphing Quantitative Data
Histogram
Boxplot
Graphing Categorical Data
Pie Chart
Bar Plot
Pie Chart
2
Misleading Graphs
How to spot a misleading graphic:
1. It may not have axis labels or _____________________.
2. It may ________________________ the x or y axis, or start at a weird place.
3. It may use _________________ for bar graphs (or a ‘pictograph’).
Example 1:
Why is the pictograph to the left misleading?
Lesson 1.1 Discussion
Discussion Question: What story does this graphic tell?
Is that story misleading? Explain.
Graphic from 2015 Planned Parenthood congressional hearing
3
Lesson 1.1 Practice
Misleading Data Graphics
A
Graphic ‘A,’ which we analyzed earlier, was
prepared by the organization: Americans
United For Life. It uses data from Planned
Parenthood’s annual reports.
Graphic ‘B’ was published on the White
House’s official blog during the Obama
Administration. It uses national school data
prepared by the Department of Education.
Graphic ‘C’ was presented at summit of
climate change skeptics. It uses global landocean temperature data from NASA’s
Goddard Institute for Space Studies.
B
For graphics ‘B’ and ‘C,’ answer these
questions:
-Is the visual misleading? Why or why not?
Source link: https://obamawhitehouse.archives.gov/blog/2016/10/17/graduation-rate-reaches-new-high-one-student-shares-his-story
C
4
A
A (same data, new graphic): In what
way(s) is this graphic more transparent
than the graphic on the previous page of
the handout?
Data courtesy of Politifact: https://www.politifact.com/factchecks/2015/oct/01/jason-chaffetz/chart-shown-planned-parenthood-hearing-misleading-/
B
B (same data, new graphic): In what
way(s) is this graphic more transparent
than the graphic on the previous page
of the handout?
C
C (including more data): In what way(s)
is this graphic more transparent than
the graphic on the previous page of the
handout?
5
Can Joy Smell Parkinson’s Disease?
Joy Milne participated in a study where she was given 12 t-shirts, half of which were worn by
Parkinson’s patients, and half of which were worn by a control group. Joy correctly identified 11 out of
the 12 shirts. Does this provide convincing evidence that Joy can smell Parkinson’s?
1. Why would it be important to know that someone can smell Parkinson’s disease?
2. How many correct decisions would you expect Joy to get out of 12 if she really couldn’t smell
Parkinson’s (she was just guessing)? Explain.
3. Do we have some evidence that Joy can smell Parkinson’s? Why?
4. How many correct decisions out of 12 would it take to convince you that Joy really could smell
Parkinson’s?
Thank you to AP Stats teacher Doug Tyson for this awesome lesson!
6
Let’s investigate whether Joy’s result could have happened purely by chance, just by guessing.
Working in pairs, you will simulate this study. One person will be the experimenter and one person will
be Joy and then you will switch.
Important: the experimenter should not reveal the truth for each shirt. They should simply record
whether the guess was correct or incorrect.
4. As the experimenter, keep track of the results:
Correct
Incorrect
5. Count up the number of correct decisions. Write the number on a sticker dot and bring it to the
poster at the front of the room. Copy the dotplot here.
‘
0
2
4
8
6
10
12
6. What does each dot represent?
7. Based on the class simulation, what proportion of the simulations resulted in 11 or more correct
identifications?
8. Based on these results, do we have convincing evidence that Joy can smell Parkinson’s? Explain.
Thank you to AP Stats teacher Doug Tyson for this awesome lesson!
7
AP Statistics Handout: Lesson 1.2
Topics: marginal & conditional distributions, barplots, segmented barplots, associations
Lesson 1.2 Guided Notes
Stop and Frisk Data
This 2011 data is a random sample of police stops from
New York’s “Stop and Frisk” program. The program
allowed police officers to stop people on the street and
search them for weapons or contraband. The program
was controversial. Critics alleged that it led to
heightened police discrimination of people of color.
Force Level
No Force Used
Hands Used
Higher Force Level*
Race of Suspect
Black
Hispanic White
1371
853
260
263
188
30
109
72
18
*Includes push to wall/ground, handcuffs, draw/point weapon, pepper spray, baton
Data source: NYC.GOV, https://www1.nyc.gov/site/nypd/stats/reports-analysis/stopfrisk.page
Two-Way Table: A table of counts describing two ________________________. Each _______________
represents one variable.
Ex: -Force Level (categorical)
-Race (categorical)
Marginal & Conditional Distributions
Marginal distributions
i) What number is always used when calculating marginal distributions?
Force level
ii) Find the marginal distribution for force level:
Proportion
No Force Used
Hands Used
Higher Force Level
Race
iii) Find the marginal distribution for race:
Black
Hispanic
White
8
Proportion
Conditional distributions
i) Find the conditional distribution for each race:
Black
White
No Force Used
Hands Used
Higher Force Level
Is this a good take? A television commentator says, “Police had ‘no force’ interactions with only 260
white suspects. Meanwhile, a much higher number of black suspects—1,371—didn’t experience force.
Clearly, black people experience ‘force-free’ interactions with police more frequently than white
people.”
Was that a good analysis of this data? Why or why not?
Barplots
What slogan should we always
remember when graphing?
9
Segmented Barplots
Please draw your representation of the segmented barplot below:
Associations
i) What three components do you need when writing an answer for an association question?
ii) Does the data suggest there is an association between race of suspect and level of force used?
There appears to be _______________________ between race and level of force used. For example,
white suspects receive no force from police officers at ____________________ (84.4%) than black
(78.7%) and Hispanic (76.6%) suspects. So, level of force is associated with race of the suspect.
Lesson 1.2 Discussion
Force level
Black
Hispanic
White
No Force
78.7%
76.6%
84.4%
Hands Used
15.1%
16.9%
9.7%
Higher Force
6.3%
6.5%
5.8%
Discussion Question: Does our data provide enough
evidence to prove that New York police are racially
biased (in terms of use of force)? Why or why not?
10
1.1 PRACTICE!
1. What variables are measured? Identify each as categorical or quantitative. In what units were the quantitative
variables measured?
State
Number of Family Members
Age
Gender
Kentucky
Florida
Wisconsin
California
Michigan
Virginia
Pennsylvania
Virginia
California
New York
2
6
2
4
3
3
4
4
1
4
61
27
27
33
49
26
44
22
30
34
Female
Female
Male
Female
Female
Female
Male
Male
Male
Female
Marital
Status
Married
Married
Married
Married
Married
Married
Married
Never married/ single
Never married/ single
Separated
Total Income
Travel time to work
21000
21300
30000
26000
15100
25000
43000
3000
40000
30000
20
20
5
10
25
15
10
0
15
40
2. A sample of 200 children from the United Kingdom ages 9-17 was selected from the CensusAtSchool website
(www.censusatschool.com). The gender of each student was recorded along with which super power they would most
like to have: invisibility, super strength, telepathy (ability to read minds), ability to fly, or ability to freeze time. Here are
the results:
Invisibility
Super Strength
Telepathy
Fly
Freeze Time
Total
Female Male Total
17
13
30
3
17
20
39
5
44
36
18
54
20
32
52
115
85
200
a. What proportion of males want the power of invisibility?
b. What proportion of females want the power of freeze time?
c. What proportion of children that want the power of telepathy are male?
d. What proportion of children that want the power of fly are female?
11
Female Male Total
Invisibility
17
13
30
Super Strength
3
17
20
Telepathy
39
5
44
Fly
36
18
54
Freeze Time
20
32
52
Total
115
85
200
3. Create a well labeled segmented bar graph of the distributions of power preference and gender. Be sure
to include a key.
Female:
Males:
Key:
4. Based on the graphs above, can we conclude that boys and girls differ in their preference of superpower? Give
appropriate evidence to support your answer.
12
How are your favorite classes related?
Is your favorite elective class associated with your favorite core class? Collect class
data to see if there is a relationship.
1. Which of the following is your favorite elective class? You must choose only one
and mark your choice on the board.
Art
Music
Physical
Education
Foreign
Language
Technology
2. Identify the individuals and variable?
3. Is the variable categorical or quantitative?
4. Go to stapplet.com to enter the class data. Make a bar graph and a pie chart.
Sketch them below.
5. Sometimes it is helpful to investigate more than one variable. Come to the board
and put a tally mark where you belong.
Find each of the following:
Core Class
% of all students who chose P.E.:
Math
English
Art
Music
Elective
% of all students who chose Math and
chose Art:
P.E.
Foreign Lang.
Tech.
% of the students who prefer math that
chose Tech.
13
6. How many variables does the table have? Are the variables categorical or
quantitative?
7. Which variable would best explain or predict the other variable?
8. Go to stapplet.com and enter the data. Make a side-by-side bar graph and a
segmented bar graph. Sketch them below.
9. How do the bars in the side-by-side-bar graph relate to the bars in the
segmented bar graph?
10. Is there an association between favorite core subject and favorite elective? If so,
describe it.
11. If there was not an association between favorite core subject and favorite
elective, what would the graphs look like? Explain.
14
Analyzing Categorical Data
Important Ideas:
Check Your Understanding:
1. The following graph was displayed by a
national news organization. Explain why
the graph may be misleading, and
sketch a corrected version of the graph.
2. A real estate agent is collecting data on the number of houses built in his town’s
three neighborhoods during three different decades. The table below gives
information.
1960s
1970s
1980s
Shady Lane
40
30
10
Oakcrest
60
15
5
Pinewood Estates
0
45
15
a. What proportion of the houses shown were built in Pinewood Estates?
b. Find the distribution of Decade Built for the houses in this town using relative
frequencies.
c. What percent of homes were built in Oakcrest and in the 1960s?
15
What will be the mascot?
When the high schools were built , the schools needed to pick a mascot. The
principal decided to have the students and teachers vote between three choices:
thoroughbreds, bumble bees, or bull dogs. His random sample results were:
Teachers
Students
Thoroughbreds Bees
80%
10%
30%
60%
Bull Dogs
10%
10%
1. Create two bar graphs below to display the results. Use three different colors for the bars.
2. Complete the third graph by taking each bar from the teacher sample and stacking them.
Use the colors to mark each section. Do the same for the student sample.
Teachers
Students
3. According to your displays, which mascot appears to have the most support? Explain.
4. Upon hearing the results of the surveys, the students argued that the decision was
incorrect because 100 teachers had been surveyed and 500 students had been
surveyed. Use this information to fill in the table below with the number of responses.
Rams
Falcons
Prairie Dogs
Teachers
Students
5. How many times more students were sampled than teachers? _____. How can you
update the third graph in #1 to take into account the sample size? Adjust your graph.
6. What should they make the mascot? Explain.
16
Representing Two Categorical Variables
Important Ideas:
Check Your Understanding:
The following table gives the result of a random sample of upper level students at Rocky Vista
University (the Fighting Prairie Dogs!), along with a mosaic plot.
Employment Status
Currently working
Not working but have had a job
Never had a job
Grade Level
Junior
14
22
15
Senior
30
40
10
a. Calculate the proportion of Juniors that are currently working, not working but have had a
job, and never had a job.
b. Calculate the proportion of Seniors that are currently working, not working but have had a
job, and never had a job.
c. Write a few sentences summarizing what the display in part (a) reveals about the
association between grade level and job experience for the students in the sample.
17
AP Statistics Handout: Lesson 1.3
Topics: relative frequency, dot plots, stemplots, histograms, CSOCS
Lesson 1.3 Guided Notes
Relative Frequency
Salaries
(thousands of $)
Salary
Frequency
Relative Frequency
29
1
8.3%
32
1
8.3%
34
4
33.3%
35
2
16.7%
39
1
8.3%
43
1
8.3%
Frequency Table
Raw Data
39
34
34
35
34
32
43
34
185
35
29
67
67
1
8.3%
185
1
8.3%
Total
12
100%
Dot Plots
What are some advantages
to using a dot plot? What
are some disadvantages?
Title
Tick, Tick
Label, Label, Label
Stemplots
Stems
Title
Tick, Tick
Label, Label, Label
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
9
24444559
3
2
3
4
5
6
…
18
7
Leaves
Key:
3|2 represents a
worker with a salary
of $32,000
5
18
9
24444559
3
7
5
Key:
3|2 represents a
worker with a salary
of $32,000
What is wrong about the above stemplot?
Why is this problem important?
Salary
29
32
34
35
39
43
67
185
Total
Frequency
1
1
4
2
1
1
1
1
12
Relative Frequency
8.3%
8.3%
33.3%
16.7%
8.3%
8.3%
8.3%
8.3%
100%
Grouped Data Values
Individual Data Values
Histograms
Salary
20-29
30-39
40-49
50-59
60-69
…
180-189
Total
Frequency
1
8
1
0
1
Relative Frequency
8.3%
66.7%
8.3%
0%
8.3%
1
12
8.3%
100%
When is it useful to group data values?
Relative Frequency Histogram
Histogram
Title
Tick, Tick
Label, Label, Label
CSOCS
Shape
“Describe the distribution…”
Context – What ________________ is being measured?
Unimodal
Shape – ___________ skew, symmetric, modes
Outliers – _____________ points
___________
_
Center – Mean, median, _____________________
Spread – ____________, IQR, standard deviation
__________
19
Symmetric
_________
i) Describe the distribution in the space below.
Label each part of CSOCS:
Describing the shape of stemplots
20
1.2 PRACTICE!
Smart Phone Battery Life
Smart Phone
Battery Life (minutes)
Apple iPhone
300
Motorola Droid
385
Palm Pre
300
Blackberry Bold
360
Blackberry Storm
330
Motorola Cliq
360
Samsung Moment
330
Blackberry Tour
300
HTC Droid
460
Here is a dotplot
Collection
1 of the data:
300
Dot Plot
340 380 420 460
BatteryLife (minutes)
1. Describe the shape, center, and spread of the distribution. Are there any (potential) outliers?
Dotplot of EnergyCost vs Type
Type
Top vs. Bottom Freezers
How do the annual energy costs (in dollars) compare for refrigerators with top freezers and refrigerators with
bottom freezers? The data below is from the May 2015 issue of Consumer Reports.
bottom
top
56
70
84
98
112
EnergyCost
126
140
2. Compare the distributions of annual energy costs for these two types of refrigerators.
21
Who’s Taller?
Which gender is taller? A sample of 14-year-olds from the United Kingdom was randomly selected using the
CensusAtSchool website.
Here are the heights of the students (in cm):
Male: 154, 157, 187, 163, 167, 159, 169, 162, 176, 177, 151, 175, 174, 165, 165, 183, 180
Female: 160, 169, 152, 167, 164, 163, 160, 163, 169, 157, 158, 153, 161, 165, 165, 159, 168, 153, 166, 158, 158,
166
Here is a back-to-back stemplot comparing male and female heights:
Female
332
98887
433100
99876655
Male
15 14
15 79
16 23
16 5579
17 4
17 567
18 03
18 7
Key: 15|1 represents a
student who is 151 cm tall.
3. Compare the distributions of height for females and males.
22
How many pairs of shoes do you own?
1. How many pairs of shoes do you own? Record your answer on the board.
2. Is “Number of pairs of shoes” a categorical or quantitative variable?
3. Enter the data at www.stapplet.com. Make a dotplot, stemplot, and histogram and
sketch each below.
4. List the mean and median of the distribution. Which value do you think is a more
appropriate measure of center? Explain.
5. Describe the distribution of the number of pairs of shoes for your class.
Shape:
Outliers:
Center:
Variability (spread):
6. Which of the three types of display do you prefer? Why?
23
Displaying Quantitative Data
Important Ideas:
Check Your Understanding:
1. Mr. Wilcox is a huge fan of University of Michigan football. His favorite season
was the 1997 season (a perfect season!). The dotplot shows the number of
points scored by the U of M team in the 12 games that season.
(a) Use the dotplot to create a stemplot of the distribution.
(b) Describe the shape of the distribution.
(c) Are there any potential outliers? Why?
(d) What measure of center is most appropriate to describe the distribution? Explain.
24
AP Statistics Handout: Lesson 1.4
Topics: measures of center, measures of spread, using technology to find summary stats
Lesson 1.4 Guided Notes
Measures of Center
Mean:
𝑥̅ =
𝑥̅ =
=
𝑛
∑ 𝑥𝑖
𝑛
39 + 34 + 34 + 35 + 34 + 32 + 43 + 34 + 185 + 35 + 29 + 67 601
=
= _______
12
12
Median:
Show the steps to finding the median:
29, 32, 34, 34, 34, 34, 35, 35, 39, 43, 67, 185
Salaries
(thousands of $)
39
34
34
35
34
32
43
34
185
35
29
67
Question Preview (for later discussion): The boss is trying to hire you to work at this company. She says,
“Our typical salary is $50,100.” Is this misleading? Why or why not?
Approximating a median in a histogram: The graph below describes 55 ACT scores.
1. Divide the sample size (the total number of
data points) by 2. This is the ______________
of the ____________ value of the dataset –
the median.
2. From the lowest value datapoints, count
the ________________ until you reach the
central position from step #1.
3. Report the _____________ of possible
values for the median.
25
Measures of Spread
Range:
Range = Max – Min
Range = 185 – 29 = _____
Standard Deviation:
𝒔𝒙 = 𝟒𝟑. 𝟔𝟏
The salaries in the dataset are typically
$43,610 _______________ the mean.
𝑠𝑥 = √
∑(𝑥𝑖 − 𝑥̅ )2
𝑛−1
Interquartile Range (IQR):
Show the steps to finding the interquartile range (IQR):
Formula: IQR = Q3 – Q1
29, 32, 34, 34, 34, 34, 35, 35, 39, 43, 67, 185
Question Preview (for later discussion): Which measure of spread (range, standard dev., or IQR) best
represents the “typical” distance between salaries? Why?
26
Technology: Summary Statistics
1. Put data into List 1
(STAT → EDIT)
2. Find 1-Var Stats
(STAT → CALC → 2)
3. Select Data List
(→ Calculate)
4. Scroll through
the summary stats
(use Sx for stdev.)
Lesson 1.4 Discussion
1. The boss is trying to hire you to work at this company. She says,
“Our typical salary is $50,100.” Is this misleading? Why or why not?
Measures of Center
Mean: 50.1 ($50,100)
Median: 34.5 ($34,500)
2. Which measure of spread (range, standard dev., or IQR) best
represents the “typical” distance between salaries? Why?
Measures of Spread
Range: $156,000
𝑆𝑥 : $43,610
IQR: $7,000
27
Salaries
(thousands of $)
39
34
34
35
34
32
43
34
185
35
29
67
“Resistance is futile”
The median is ________________ (not seriously affected by) skew and outliers. The mean
_______________________ to skew and outliers. The mean follows skew/outliers.
The interquartile range (IQR) is resistant to skew and outliers. The range and ______________________
are not resistant to skew and outliers.
Why are the median and IQR resistant to outliers? Let’s explore with the salary data:
29, 32, 34, 34, 34, 34, 35, 35, 39, 43, 67, 185
For the mean: The outlier salary – $185,000 – drags up the mean because its high value is given _______
___________ in the calculation
For the median: The ______________ matters more than the __________. Because $185,000 is the
highest data point, it’s crossed off right away. The outlier is _____ given a large weight in the calculation.
For the IQR: Like the median, the position matters more than the value for the IQR.
Applet: Use the (very cool) simulation linked here to explore these properties of the mean and median.
Link: http://digitalfirst.bfwpub.com/stats_applet/stats_applet_6_meanmed.html
Source: Digital First project from Bedford, Freeman, & Worth publishers
Right Skew
Symmetric
Left Skew
Med. ____ Mean
Med. ____ Mean
Med. ____ Mean
28
How many colleges are you applying to?
How many different colleges is your group of 4 applying to? Find the total number of
colleges for your whole group.
1. Record the data for the class here.
2. Calculate the mean and median for the set of data. Compare them.
3. What is the range of the data?
Finding Standard Deviation
4. Finding range is helpful but it
does not tell us how spread out the
data is between the minimum and
maximum. How can we find the
average distance of the values
from the mean?
Value
Distance from mean
(Distance from mean)2
a. Complete the table.
b. The average you calculated is the
average of the squared distances
from the mean. How do we use this
to find the average distance from
the mean? Find it.
Total:
Average (Distance from mean)2:
5. Go to stapplet.com. Enter the classroom data and find the summary statistics. Verify
our work. How does it compare?
6. We forgot to add one group that applied to 40 colleges! Add this group to the data
set. Calculate the new mean, median and standard deviation using the applet. How
does it compare to the original measures? Why do you think this is?
29
Describing Quantitative Data
Important Ideas:
Check Your Understanding:
A researcher is interested in how much annual rainfall is typical in the United States. She
takes a random sample of 9 cities in the U.S. and records the annual rainfall, in inches.
8.2
10.3
33.5
39.1
40.5
41.9
42.4
44.9
53.7
1. Calculate the mean annual rainfall for these cities.
2. Find the median annual rainfall for these cities.
3. Would you use the mean or the median to summarize the typical annual rainfall
for a U.S. city? Explain.
4. The standard deviation of the annual rainfall for these 9 cities is 15.52 inches.
Interpret this value.
30
AP Statistics Handout: Lesson 1.5
Topics: five-number summary, determining outliers, boxplots, comparing distributions
Lesson 1.5 Guided Notes
Five-Number Summary
Determining Outliers
Formulas:
Outliers are __________
high or low data values.
Upper Limit: Q3 + 1.5 x IQR
Lower Limit: Q1 – 1.5 x IQR
Example:
Upper Limit: 41 + 1.5 x 7 = 51.5
Lower Limit: 34 – 1.5 x 7 = 23.5
Which of the following salaries are outliers?
Data values ______
this are outliers.
29, 32, 34, 34, 34, 34, 35, 35, 39, 43, 67, 185
Data values _______
this are outliers.
Boxplots
31
_________
Right Skew
a) What percent of the data is below Q1?
b) What percent of the data is below Q3?
c) What percent of the data is above Q1?
d) What percent of the data is above the median?
e) What percent of the data is within the IQR?
Describe the distribution…
Context: Subject of data
Shape: Skew (not modes)
Outlier: _________  for boxplots
Center: _____________  for boxplots
Spread: ______ for boxplots
Data was collected on ___________ at a company. The distribution
appears ______________ (high outliers). There are two high outliers, at
____________________. The ______________ of the distribution is
$34,500. The ______ is $7,000.
32
Comparing Distributions
Two tax plans (A & B), each with average tax cuts of $4,000
1. If comparing dotplots, stemplots, histograms, or boxplots…
- use _________
2. Use ________________ language for each feature…
- “less than,” “greater than,” “similar,” etc.
C (Context) – tax cuts ($) under two plans
S (Shape) – A is left skew, B is right skew
O (Outlier) – A has no outliers, B has high outliers
C (Center) – Median A is 4.75, Median B is 0.20
S (Spread) – IQR A is 2.75, IQR B is 0.60
Household ____________ under Plan A are left skew,
whereas they are severely right skew under Plan B. There
are no outliers under Plan A, while there are ________
_____________ higher than $12,000 in Plan B. The median
tax cut under Plan A (about $4,750) is __________ than
under Plan B (about $200). Plan A also has a _________ IQR
(about $2,750) than Plan B (about $600).
Lesson 1.5 Discussion
Discussion Question: Answer the Press Secretary’s
question – what reason could a congressperson give
for opposing a tax plan that produces an average
tax cut of $4,000?
From official White House Press Secretary Twitter account:
https://twitter.com/presssec/status/922245672409198597?lang=en
33
Critics:
Low-Income: no tax cut
Middle Income: no tax cut
High Income: large tax cut
Averages ________ skew and outliers – this is
sometimes called the “flaw of averages”
= Large ___________ tax cut
Critics of the tax plan argue that, once we get reliable data, the Trump tax cut will look more like Plan B
than Plan A. The median tax cut will be much lower than $4,000, with tax cuts well below $1,000 for the
majority of households (look at the IQR).
This is why we describe the _____________________ of data – a one number summary can be
deceiving.
Image courtesy of Sam Savage and Jeff Danziger from The Flaw of Averages
34
1.3 PRACTICE!
McDonald’s Beef Sandwiches
Here are data for the amount of fat (in grams) for McDonald’s beef
sandwiches:
Sandwich
Hamburger
Cheeseburger
Double Cheeseburger
McDouble
Quarter Pounder®
Quarter Pounder® with Cheese
Double Quarter Pounder® with Cheese
Big Mac®
Big N' Tasty®
Big N' Tasty® with Cheese
Angus Bacon & Cheese
Angus Deluxe
Angus Mushroom & Swiss
McRib ®
Mac Snack Wrap
Fat (g)
9g
12 g
23 g
19 g
19 g
26 g
42 g
29 g
24 g
28 g
39 g
39 g
40 g
26 g
19 g
Use your graphing calculator to find the following:
Mean
Median
5 Number Summary
IQR
Are there any outlier/s
using the IQR*1.5 Rule?
The Previous Home Run King
Using your graphing calculator, create a box plot using the data below. Be sure to identify each number in a five number
summary and any outliers using the IQR*1.5 Rule.
Number of home runs that Hank Aaron hit in each of his 23 seasons:
13 27 26 44 30 39 40 34 45 44 24 32 44 39 29 44 38 47 34 40 20 12 10
35
Who Has More Contacts—Males or Females?
The following data show the number of contacts that a sample of high school students had in their cell phones. Do the
data give convincing statistical evidence that one gender has more contacts than the other? You need both graphical
and numerical evidence.
Male: 124 41 29 27 44 87 85 260 290 31 168 169 167 214 135 114 105 103 96 144
Female: 30 83 116 22 173 155 134 180 124 33 213 218 183 110
36
Where Do I Stand?
How does my height compare with the other AP Stats students in my class? In order
to answer this question, Ashmita, a student in 4th hour AP Stats, recorded the
heights of everyone in her class. The heights (in inches) were:
68 72 61 62 63 63 64 64 59 62 61 60 65 62 57 77 62 71 65 62 70
1. Create a dotplot to display the class distribution of heights.
2. What is the median height? Describe how you found it.
3. What is Q1 and Q3? Describe how you found them.
4. Record the following values and then use them to make a boxplot.
Minimum:
Q1:
Median:
Q3 :
Maximum:
4. The interquartile range (or IQR) is defined as Q3 − Q1. Find the IQR. Where do you see
the IQR in the boxplot?
5. An outlier is a data value that is way too small or way too big (using the rules below). Are
there any outliers? Show your work.
Way too small < Q1 −1.5IQR
Way too big > Q3 +1.5IQR
6. Ashmita is 63 inches tall. How does her height compare with the other AP Stats students
in her class?
37
Describing Quantitative Data
Important Ideas:
Check Your Understanding:
Mr. Wilcox is a huge fan of University of Michigan football. His favorite season was the 1997
season (a perfect season!). Here is a back-to-back stemplot of the points scored by the 1997
University of Michigan football team and the archrival Michigan State University football
team. Write a few sentences comparing the distributions.
38
Who is Baseball’s Greatest Home Run Hitter?
Barry Bonds broke Mark McGwire’s record when he hit 73 home runs in the 2001 season. How
does this accomplishment fit with the rest of Bond’s career? Here are Bond’s home run counts
for the years 1986 to 2007.
16 25 24 19 33 25 34 46 37 34 49 73 46 45 45 5 26 28
1. What display did your group get assigned? _____________________
2. Create the display on the whiteboard and bring it to the front of the room.
3. Describe the distribution.
4. What are the advantages and disadvantages of your type of display?
5. Below is this distribution of the number of home runs per season for Mark McGwire.
Compare this distribution to the one for Barry Bonds.
39
STATISTICS
SECTION II
Part A
Questions 1-5
Spend about 1 hour and 5 minutes on this part of the exam.
Percent of Section II score—75
Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
1. The sizes, in square feet, of the 20 rooms in a student residence hall at a certain university are summarized in the
following histogram.
(a) Based on the histogram, write a few sentences describing the distribution of room size in the residence hall.
(b) Summary statistics for the sizes are given in the following table.
Mean
Standard
Deviation
Min
Q1
Median
Q3
Max
231.4
68.12
134
174
253.5
292
315
-640
GO ON TO THE NEXT PAGE.
Determine whether there are potential outliers in the data. Then use the following grid to sketch a boxplot of
room size.
(c) What characteristic of the shape of the distribution of room size is apparent from the histogram but not from
the boxplot?
41
®
2015 AP STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
Part A
Questions 1-5
Spend about 65 minutes on this part of the exam.
Percent of Section II score—75
Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
1. Two large corporations, A and B, hire many new college graduates as accountants at entry-level positions. In
2009 the starting salary for an entry-level accountant position was $36,000 a year at both corporations. At each
corporation, data were collected from 30 employees who were hired in 2009 as entry-level accountants and were
still employed at the corporation five years later. The yearly salaries of the 60 employees in 2014 are
summarized in the boxplots below.
(a) Write a few sentences comparing the distributions of the yearly salaries at the two corporations.
(b) Suppose both corporations offered you a job for $36,000 a year as an entry-level accountant.
(i) Based on the boxplots, give one reason why you might choose to accept the job at corporation A.
(ii) Based on the boxplots, give one reason why you might choose to accept the job at corporation B.
42
Unit 1
Chapter 2
Modeling
Distributions of
Quantitative Data
43
AP Statistics Handout: Lesson 2.1
Topics: percentiles, cumulative relative frequency, standardized scores (z-scores)
Lesson 2.1 Guided Notes
Percentiles
Percentile: the percent of data _____________ a certain data value.
At what percentile is the person who makes a salary of $43,000?
29, 32, 34, 34, 34, 34, 35, 35, 39, 43, 67, 185
1) What percentile is Q1?
2) What percentile is the median?
3) What percentile is Q3?
Whose score is more impressive? Why?
44
Cumulative Relative Frequency
1) Is an ACT score of 18 a good score?
2) You are applying for an elite college and want to
score in the top quartile of test takers. What score
do you need?
Standardized Scores (Z-Scores)
Z-Scores (also called standardized scores): measures how many ______________________ a data point
is __________________ the mean.
𝑧=
𝑑𝑎𝑡𝑎 𝑝𝑜𝑖𝑛𝑡 − 𝑚𝑒𝑎𝑛
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
|
z=
𝑥𝑖 − μ
𝜎
Variation Matters:
14 PPG
In which league (A or B) is this player relatively
“better?” Why?
Standardization: A point’s location in the distribution depends on both distance from the ____________
and the distribution’s _________ or _________________.
45
Standardized: Who Was the Best?
Show all z-score calculations below:
Who was the G.O.A.T?
30.1 ppg
30.1 ppg
27.1 ppg
Mean (𝜇) ppg
in their era:
10.8 ppg
8.7 ppg
8.4 ppg
Stdv (𝜎) ppg
in their era:
7.0 ppg
5.9 ppg
5.5 ppg
MJ’s PPG was ____ standard
deviations ______ the mean
for his era, making him the
most _____________ scorer
of these three legends.
Z-Score:
(show work)
Standardized: Players who were not the best…
Photo: opencourt-basketball.com
Calculate Adam Morrison’s z-score for PPG…
While with the Lakers, he averaged ______ PPG.
(League: µ = 8.4 ppg, σ = 5.5 ppg)
z=
Adam Morrison
______ − 8.4
= _________
5.5
Adam Morrison’s scoring rate was 1.1 standard deviations
_____________ the league _______________ in his era.
Positive and Negative Z-Scores
data value > mean → ______________
data value < mean → ______________
𝑧=
𝑑𝑎𝑡𝑎 𝑝𝑜𝑖𝑛𝑡 − 𝑚𝑒𝑎𝑛
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
46
Positive Z-Score: The number of standard
deviations _________ the mean.
Negative Z-Score: The number of standard
deviations _________ the mean.
2.1 PRACTICE!
1. The stemplot below shows the number of wins for each of the 30 Major League Baseball teams in 2009.
5
6
7
8
9
10
9
2455
00455589
0345667778
123557
3
Key: 5|9 represents a
team with 59 wins.
Find the percentiles for the following teams:
(a) The Colorado Rockies, who won 92 games.
(b) The New York Yankees, who won 103 games.
(c) The Kansas City Royals and Cleveland Indians, who both won 65 games.
2. Here is a table showing the distribution of median household incomes for the 50 states and the District of Columbia.
Calculate the relative frequency and cumulative relative frequency.
Median
Income
($1000s)
Frequency
35 to < 40
1
1
40 to < 45
10
11
45 to < 50
14
25
50 to < 55
12
37
55 to < 60
5
42
60 to < 65
6
48
65 to < 70
3
51
Relative
Frequency
47
Cumulative
Frequency
Cumulative
Relative
Frequency
3. Use the cumulative relative frequency graph for the state income data to answer each question.
(a) At what percentile is California, with a median income of $57,445?
(b) Estimate and interpret the first quartile of this solution.
4. Miami-Dade County Public Schools employs teachers at salaries between $40,000 and $71,000. The teachers’ union
and the school board are negotiating the form of next year’s increase in the salary schedule.
(a) If every teacher is given a flat $1000 raise, what will this do to the mean salary? To the median salary? Explain your
answers.
(b) What would a flat $1000 raise do to the extremes and quartiles of the salary distribution? To the standard deviation
of teachers’ salaries? Explain your answers.
48
Where do I stand?
How does my height compare with other AP Stats students?
The dotplot below represents a random sample of the heights of 20 AP Stats students
to the nearest inch.
1.
Describe the distribution.
2.
a. Arianna is 65 inches tall. What percent of the heights are less than or
equal to 65?
b. What is your height?
than or equal to your height?
3.
Complete the table.
What percent of the heights are less
Height Frequency
Relative
Cumulative
Frequency
Relative Freq.
56-60
61-65
66-70
71-75
76-80
4.
Use the info in the table to create a cumulative relative frequency graph.
a. Mrs. Gallas is 66 inches tall.
Estimate and interpret the percentile
she is at using the graph.
Cumulative
Relative
Frequency
b. Estimate and interpret the 80th
percentile.
Height (in)
49
Describing Location in a Distribution
Important Ideas:
Check Your Understanding:
1. According to a 2019 article at Insider.com, the state of Pennsylvania was at the 82nd percentile
for Pre-K to 12th grade education and was at the 0th percentile for higher education. Explain
what these values mean.
2. The graph displays the cumulative relative frequency of the cost of in-state public college
education for each of the 50 states.
Cumulative relative frequency (%)
120
100
80
60
40
20
0
0
2000
4000
6000
8000
10000
12000
14000
16000
In-State Public College Tuition ($)
a. About what percent of states have in-state public college tuition less than or equal to
$8000? More than $8000?
b. Estimate Q1, Q3, and the IQR of the distribution of phone in-state public tuition.
50
AP Statistics Handout: Lesson 2.2
Topics: transforming data (adding, subtracting, multiplying, and dividing data by a constant)
Lesson 2.2 Guided Notes
Effects of adding/subtracting to a distribution
Adding $6000 to each salary below (in thousands of $):
29, 32, 34, 34, 34, 34, 35, 35, 39, 43, 67, 185
+6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6 +6
Original dataset
Median = 34.5
Range = 156
35, 38, 40, 40, 40, 40, 41, 41, 45, 49, 73, 191
Transformed dataset
Median = 40.5
Range = 156
Shifted up by _____
Stayed __________
Subtract 6 from each datapoint
Add 6 to each datapoint
When adding or subtracting every data value by a constant…
1. Center: adds/subtracts by that constant amount.
2. Spread: remains the ____________.
3. Shape: remains the _____________.
51
Effects of multiplying/dividing a distribution by a constant
Simple Example: Multiply the following dataset by 3…
x3
1, 2 → 3, 6
Median: 1.5
Range: 1
x3
Median: 4.5
Range: 3
Multiplication and Spread
When multiplying, big numbers __________
______ than small numbers. Because the
numbers change at different rates, the
spread ___________________ grows larger.
6% salary increasing (multiply by 1.06):
29, 32, 34, 34, 34, 34, 35, 35, 39, 43, 67, 185
Original dataset
Median = 34.5
Range = 156
x1.06 x1.06 x1.06 x1.06 x1.06 x1.06 x1.06 x1.06 x1.06 x1.06 x1.06 x1.06
30.7, 33.9, 36.0, 36.0, 36.0, 36.0, 37.1, 37.1, 41.34, 45.6, 71.0, 196.1
Transformed dataset
Median = 36.6
Range = 165.4
Grew by a factor of 1.06
Grew by a factor of 1.06
Multiply each datapoint by 2
When multiplying or dividing every data value by a constant…
1. Center: multiplies/divides by that constant amount.
2. Spread: _____________________ by that constant amount.
3. Shape: remains the ____________.
52
Divide each datapoint by 2
Lesson 2.2 Discussion
Transformation (Multiplication) of Money: Stocks
Stock: a small ____________________ share of a company.
• If you buy a stock of Coca-Cola, you are a part owner (a tiny
part) of Coca-Cola. If the company makes profits and gains
value, your stock ___________ in value. If the company
suffers losses, your stock ___________ value.
S&P 500: an _____________ that tracks the value of stock from 500 large American companies.
• It’s a good indicator of how the stock market is performing overall.
Stocks (and other wealth assets) tend to multiply in value over time.
Example: Let’s say we invested $185 (in today’s dollars) in an S&P 500 index stock fund in 1950. How
much money would we have in 2020?
Year
1950
1951
1952
1953
1954
Value
$185.00
$192.40
$200.10
$208.10
$216.42
×1.04
×1.04
×1.04
×1.04
2016
2017
2018
2019
2020
$2,462.48
$2,560.98
$2,663.41
$2,769.95
$2,880.75
×1.04
×1.04
×1.04
×1.04
Assumptions:
-Gains 4% in value per year (inflation adjusted)
-No dividends
You’d have $2,880.75!
Inflation adjusted using CPI for March 2020 dollars.
Data from: multpl.com/inflation-adjusted-s-p-500/table/by-year
53
Wealth Inequality in the U.S.
____________ = all assets (stocks, home
value, bonds, etc.)
For the bottom 90% of families, wealth
has _____________. That’s good!
U.S. Family Real Wealth Over Time
Discussion Questions:
1. Why is wealth inequality increasing?
Hint: think about multiplication.
2. Is wealth inequality a problem? If so, what
should we do about it? If not, why not?
U.S. Family Real Wealth Over Time
Monetary values expressed in 2010 dollars.
Data Source: Appendix Table B3 of Emmanuel Saez, Gabriel Zucman, “Wealth Inequality in the United States since 1913: Evidence
from Capitalized Income Tax Data,” The Quarterly Journal of Economics, Volume 131, Issue 2, May 2016, Pages 519–578,
https://doi.org/10.1093/qje/qjw004
54
Multiplication increases ___________
Year
Value
(Low Start)
Value
(High Start)
1950
$185.00
$100,000
1951
$192.40
$104,000
1952
$200.10
$108,160
2018
$2,663.41
$1,439,683.65
2019
$2,769.95
$1,497,271.00
2020
$2,880.75
$1,557,161.84
Initial Difference:
____________
×1.04
×1.04
×1.04
×1.04
55
Final Difference:
____________
Multiplication increases spread. So
______________ multiplication
really increases spread.
How Did I Do?
How well did you do on the Chapter 1 Test? How well did you do relative to your
classmates?
Here are the results of a random sample of 20 of the Chapter 1 Tests, along with a dotplot
and summary statistics.
Test
Scores
61 65 65 73 75 77 78 78 79 80 80 80 80 81 81 88 89 93 98 99
n mean SD min Q1 med Q3 max
20 80 10 61 76 80 84.5 99
1. Biff scored a 65. What is Biff’s percentile?
2. Was Biff above or below the mean? By how many points? By how many standard
deviations?
3. Marty scored an 88. What is Marty’s percentile?
4. Was Marty above or below the mean? By how many points? By how many standard
deviations?
A z-score is defined as the number of standard deviations above or below the mean.
z=
5. Write a formula for calculating a z-score.
6. Goldie scored a 98 on the Chapter 1 Test. Find and interpret the z-score.
Bonus: Goldie was aspiring for what job?
56
7. There are two mathematical operations used when calculating a z-score:
a. First, we take each score, and ____________________ the mean (remember the mean
was 80). Fill in the table and then make a dotplot for each.
SCORE
SCORE - MEAN
61 65 65 73 75 77 78 78 79 80 80 80 80 81 81 88 89 93 98 99
Dotplot for SCORE
Dotplot for SCORE - MEAN
What happens to the shape, center, and variability when you subtract the mean from each
score?
b. Second, we take the SCORE – MEAN and ____________________by the standard
deviation (remember the standard deviation is 10). Fill in the table and then make a dotplot
for each.
SCORE - MEAN
SCORE – MEAN
SD
-19 -15 -15 -7 -5 -3 -2 -2 -1 0
Dotplot for
0
0
0
1
1
8
9 13 18 19
SCORE − MEAN
SD
What happens to the shape, center, and variability when you divide by the standard deviation for
each value?
8. Summarize: What happens to the shape, center and variability of a distribution when you
…add or subtract the same value a from each value?
…multiply or divide by the same value b from each value?
9. What is the mean and standard deviation of the distribution of z-scores? Will this be true
for any distribution of z-scores? Explain.
57
Describing Location in a Distribution
Important Ideas:
Check Your Understanding:
1. According to an article on Yahoo!news, you should change your sheets every 7 days…at
minimum. To investigate the sheet changing habits of adults, a random sample of 20 adults
reported how often they change their sheets using an anonymous survey. Here is a dotplot
and summary statistics of the results.
a. Suppose you convert the time before changing sheets from days to weeks. Describe the
shape, mean, and standard deviation of the distribution of time before changing sheets in
weeks.
b. The adults in the study are given an article explaining the health benefits that would arise
from changing their sheets more often. After reading the article each person agrees to
change their sheets one week sooner than they used to. How does the shape, center, and
variability of this distribution compare with the distribution of time in part (a)?
c. Now suppose you convert the time before changing sheets from part (b) to z-scores. What
would be the shape, mean, and standard deviation of this distribution? Explain your
answers.
58
AP Statistics Handout: Lesson 2.3
Topics: normal curves, the empirical rule (68-95-99.7 rule), race and “intelligence”
Lesson 2.3 Guided Notes
What is a normal curve?
The Normal Curve
Frequency
1. _______________ and “bell shaped”
2. The mean = ____________, both located at
the exact center
3.
Most data points are __________ the center
Data Values
Example: IQ scores are commonly thought to be normally distributed, with mean 100 pts. and standard
deviation 15 pts.
𝝁 = 𝟏𝟎𝟎
55
70
85 100 115 130 145
𝝈 = 𝟏𝟓
The Empirical Rule (a.k.a the 68-95-99.7 rule)
68% of the data is within
_____ of the mean
95% of the data is within
_____ of the mean
68%
𝝈
55
70
𝝈
99.7% of the data is within
_____ of the mean
95%
85 100 115 130 145
𝟐𝝈
55
70
𝟐𝝈
99.7%
85 100 115 130 145
𝟑𝝈
55
59
70
𝟑𝝈
85 100 115 130 145
Strategy: Normal Curve Problems
1. ______________________ curve (include mean, standard deviation, tick marks, shade
area/point in question)
2. Perform _________________ (show work)
3. Answer the question _____________________
Examples:
a) You take an IQ test and get a score of 130. Assume IQ is normally distributed with mean 100 pts.
and standard deviation 15 pts. What percentile are you at?
b) What score would you have to earn on an IQ test to be at least at the 84th percentile?
60
Lesson 2.3 Discussion
Race and “intelligence”
-In The Bell Curve, the authors
state that black people score
about 15 points lower than
white people on IQ tests, on
average.
An argument the book makes…
On average:
• Black test takers showed below average IQ
• White test takers showed above average IQ
-Using this evidence, some
make the argument that
white people tend to be more
intelligent than black people.
Herrnstein, Richard J., and Charles
A. Murray. The Bell Curve:
Intelligence and Class Structure in
American Life. Free Press, 1994.
15
Average black
IQ scores
Average white
IQ scores
Simulated questions* from the AFQT (the test used by The Bell Curve to obtain IQ data):
Word Knowledge
1. “Solitary” most nearly means
a) sunny
b) being alone
c) playing games
d) soulful
Math Knowledge
2. In the drawing below, JK is the median of the
trapezoid. All of the following are true EXCEPT
a) LJ = JN
b) a = b
c) JL = KM
d) a ≠ c
Discussion Question: Do you believe, based on these sample questions, the AFQT accurately measures
IQ? Why or why not?
*Source: Bock, R. Darrel, and Elsie G.J. Moore. Advantage and Disadvantage: A Profile of American Youth. Lawrence Erlbaum
Associates, 1986. pg. 28-30
61
2.2 PRACTICE!
Wins in Major League Baseball
1. In 2009, the mean number of wins was 81 with a standard deviation of 11.4 wins.
Find and interpret the z-scores for the following teams.
(a) The New York Yankees, with 103 wins.
(b) The New York Mets, with 70 wins.
Batting Averages
2. In the previous alternate example about batting averages for Major League Baseball players in 2009, the mean of the
432 batting averages was 0.261 with a standard deviation of 0.034. Suppose that the distribution is exactly Normal with
 = 0.261 and  = 0.034.
(a) Sketch a Normal density curve for this distribution of batting averages. Label the points that are 1, 2, and 3 standard
deviations from the mean.
(b) What percent of the batting averages are above 0.329? Show your work.
(c) What percent of the batting averages are between 0.227 and .295? Show your work.
62
Exploring Density Curves
Complete each of the following experiments and submit your answers using the google
form. Resubmit your answers for a total of 3 submissions. Predict (sketch) what the
graphs of the class data from each experiment will look like if we did this many many
times. Draw and label lines where you predict the mean and median will be.
Experiment 1: Roll a die and record the value it lands on. 1st roll:
Prediction:
2nd roll:
3rd roll:
Actual:
Experiment 2: Try to toss a penny and make it land on the target. Measure the
distance of the penny from the target in cm. Round to the nearest tenth.
2nd Attempt:
3rd Attempt:
1st Attempt:
Prediction:
Actual:
Experiment 3: Try to stop your stopwatch at exactly 5 seconds. Record what the
stopwatch reads below. Record to the hundredths place.
2nd Attempt:
3rd Attempt:
1st Attempt:
Prediction:
Actual:
Normal Curves: Label the values 1, 2, and 3 standard deviations above and below the
mean using the stopwatch data.
What percentage of the data is within two standard
deviations of the mean?
What percentage of the data is further than two standard
deviations from the mean?
What percentage of the data is greater than 1 standard
deviation above the mean?
What percentage of the data is between z = –1 and z = 2?
63
Density Curves and Normal Distributions
Important Ideas:
Check Your Understanding:
1.
A game that is sometimes played at baby showers asks the guests to cut a length of yarn
that they believe will best measure the distance around the mom’s baby bump. At a particular
baby shower the length of string cut by the guests was uniformly distributed over the interval
40 to 50 inches.
a. What height must the density curve have? Justify your
answer.
b. About what percent of the guests cut their yarn longer
than 48 inches?
c. Calculate and interpret the 25th percentile of this distribution.
2.
The distribution of waist circumference for women who are 8 months pregnant is approximately
Normal with mean µ = 44 inches and standard deviation σ = 4 inches.
a. Sketch the Normal curve that approximates the distribution of waist circumference for
women who are 8 months pregnant. Label the mean and the points that are 1, 2, and 3
standard deviations from the mean.
b. About what percent of women who are 8 months pregnant have a waist circumference
that is less than 40 inches? Show your work.
c. A waist circumference greater than 52 inches may indicate excess amniotic fluid. What
percent of women who are 8 months pregnant may have excess amniotic fluid? Justify
your answer.
64
AP Statistics Handout: Lesson 2.4
Topics: normal calculations – finding percentages from values and values from percentages
Lesson 2.4 Guided Notes
Finding percentages from values
When between standard deviations, you must use a ____________________:
1. When finding percentages from values, use __________________ calculator command.
2. When finding values from percentages, use ___________________ calculator command.
a) You take an IQ test and get a score of 124. Assume IQ scores are normally distributed with a mean of
100 points and a standard deviation of 15 points. What percentile are you at?
Calculator Steps
2nd→VARS→ 2:normalcdf
Finding values from percentages
b) What score would you have to earn on an IQ test to be below 88% of test-takers? Assume IQ scores
are normally distributed with a mean of 100 points and a standard deviation of 15 points.
Calculator Steps
2nd→VARS→ 3:invNorm
65
3. According to the CDC, the heights of 3 year old females are approximately Normally distributed with a mean of 94.5
cm and a standard deviation of 4 cm. Be sure to draw curves for each calculation!
(a) What is the third quartile of this distribution?
(b) What is the median of this distribution?
(c) If a 3 year old female was 91.7 cm tall, what percentile would she be in?
(d) If a mother knew her daughter was at the 91st percentile in height, how tall is her daughter?
(e) If a 3 year old female was 96.4 cm tall, what percentile would she be in?
4. Scores on the Wechsler Adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately
Normally distributed with μ = 110 and σ = 25. For each part, follow the four-step process.
(a) At what percentile is an IQ score of 150?
(b) What percent of people aged 20 to 34 have IQs between 125 and 150?
(c) MENSA is an elite organization that admits as members people who score in the top 2% on IQ tests. What score on
the Wechsler Adult Intelligence Scale would an individual have to earn to qualify for MENSA membership?
66
Key points and the “genius” example
Key Points
1. Three steps: Draw+_________, Calculate, Answer
2. Percentile is the percent ___________ a data value.
3. “At least” means the data value __________________.
4. invNorm’s input of “area” is always the area _____________ the data value.
5. To approximate an upper bound of ∞, use a very large ______________ number
(such as: 1000000000)
6. To approximate lower bound of -∞, use high magnitude _______________ number
(such as: -1000000000)
Mensa International is an organization that “accepts individuals who
score in the top 2%, ie, two SDs or more above the average” on IQ
tests. Source: https://www.mensa.org/iq/what-iq, for Mensa
international, qualifications may vary by country.
a) For an IQ test with mean of 100 and stdev. of 15, what score would
you have to get to qualify for Mensa International?
b) There are about 328 million people in the United States. How many Mensa International “geniuses”
live in the United States?
Lesson 2.4 Discussion
Discussion Question: Can we identify “intelligence” using data? Is it possible to get an accurate, singlenumber summary of someone’s intellect?
67
Will Marty Make it Back to the Future?
After accelerating for 20 seconds, a DeLorean sports car has a wide range of speeds that it
can achieve, depending on traction. The distribution of speed follows an approximately
Normal distribution with a mean of 80 mph and a standard deviation of 7.7 mph.
1. Label the appropriate values
on the normal distribution
2. What percentage of the runs will give the Delorean a speed greater than 87.7 mph?
3. What percentage of the runs will give the Delorean a speed between 64.6 mph and 87.7
mph?
4. What percentage of the runs will give the Delorean a speed less than 64.6 mph?
5. What percentage of the runs will give the Delorean a speed less than 68.45 mph?
68
6. What percentage of the runs will give the Delorean a speed greater than 85 mph? Show
work.
7. What percentage of the runs will give the Delorean a speed between 70 and 95 mph?
Show work.
8. Marty wants his last run to be in the top 15% of all the possible speeds. What speed does
he need to achieve to be in the top 15%?
69
Density Curves and Normal Distributions
Important Ideas:
Check Your Understanding:
According to an article at africageographic.com there are many unexpected uses for elephant
waste. But how much waste do elephants produce per day? Studies show that the distribution of
amount of waste produced by adult elephants can be modeled by a Normal distribution with mean
250 pounds and standard deviation 21 pounds.
a.
What percent of adult elephants produce at least 300 pounds of waste in a day?
b.
A zoo has an agreement with the local farmers. They sell the daily waste from their elephant
to the farmers for $5.00 per pound. If the amount of waste produced by the elephant on a
given day is at the 10th percentile of the distribution of waste, how much money would the
zoo make selling the waste that day?
70
Do we have Normal test scores?
Here are the Chapter 1 Test scores for 50 of the current AP Statistics students
89
89
78
85
83
96
63
78
94
78
89
87
72
74
89
76
72
81
81
85
75
85
72
100
74
83
61
81
74
55
76
91
76
80
79
67
57
76
78
93
91
83
72
67
96
85
93
70
76
83
Is the distribution of Chapter 1 Test scores approximately normal? Justify your answer using
several different approaches. The group with the most convincing argument will win a prize.
71
Density Curves and Normal Distributions
Important Ideas:
Check Your Understanding:
The following figure is a Normal probability plot of the difference between the desired thickness of
a cell phone screen protector and the actual thickness of the cell phone screen protector for a
random sample of 300 screen protectors, in micrometers. Use the graph to determine if this
distribution of difference in thickness is approximately Normal.
Difference in thickness (micrometers)
72
2011 AP® STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
Part A
Questions 1-5
Spend about 65 minutes on this part of the exam.
Percent of Section II score—75
Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
1. A professional sports team evaluates potential players for a certain position based on two main characteristics,
speed and strength.
(a) Speed is measured by the time required to run a distance of 40 yards, with smaller times indicating
more desirable (faster) speeds. From previous speed data for all players in this position, the times to run
40 yards have a mean of 4.60 seconds and a standard deviation of 0.15 seconds, with a minimum time of
4.40 seconds, as shown in the table below.
Mean
Time to run 40 yards 4.60 seconds
Standard Deviation
Minimum
0.15 seconds
4.40 seconds
Based on the relationship between the mean, standard deviation, and minimum time, is it reasonable to
believe that the distribution of 40-yard running times is approximately normal? Explain.
(b) Strength is measured by the amount of weight lifted, with more weight indicating more desirable (greater)
strength. From previous strength data for all players in this position, the amount of weight lifted has a mean
of 310 pounds and a standard deviation of 25 pounds, as shown in the table below.
Mean
Amount of weight lifted 310 pounds
Standard Deviation
25 pounds
Calculate and interpret the z-score for a player in this position who can lift a weight of 370 pounds.
(c) The characteristics of speed and strength are considered to be of equal importance to the team in selecting a
player for the position. Based on the information about the means and standard deviations of the speed and
strength data for all players and the measurements listed in the table below for Players A and B, which
player should the team select if the team can only select one of the two players? Justify your answer.
Player A
Time to run 40 yards
Player B
4.42 seconds 4.57 seconds
Amount of weight lifted 370 pounds
73
375 pounds
2009 AP® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
STATISTICS
SECTION II
Part A
Questions 1-5
Spend about 65 minutes on this part of the exam.
Percent of Section II score—75
Directions: Show all your work. Indicate clearly the methods you use, because you will be graded on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
1. As gasoline prices have increased in recent years, many drivers have expressed concern about the taxes they pay
on gasoline for their cars. In the United States, gasoline taxes are imposed by both the federal government and
by individual states. The boxplot below shows the distribution of the state gasoline taxes, in cents per gallon,
for all 50 states on January 1, 2006.
(a) Based on the boxplot, what are the approximate values of the median and the interquartile range of the
distribution of state gasoline taxes, in cents per gallon? Mark and label the boxplot to indicate how you
found the approximated values.
(b) The federal tax imposed on gasoline was 18.4 cents per gallon at the time the state taxes were in effect.
The federal gasoline tax was added to the state gasoline tax for each state to create a new distribution of
combined gasoline taxes. What are approximate values, in cents per gallon, of the median and interquartile
range of the new distribution of combined gasoline taxes? Justify your answer.
© 2009 The College Board. All rights reserved.
Visit the College Board on the Web: www.collegeboard.com.
GO ON TO THE NEXT PAGE.
-674
Unit 2
Chapter 3
Exploring TwoVariable Quantitative
Data
75
AP Statistics Handout
Topics: explanatory/response, describing scatterplots, correlation coefficient (r), causation
Lesson 3.1 Guided Notes
Explanatory and response variables
Bivariate data: data with _________ variables. Two quantitative variables are visualized in a ______________
Income and Food Access Example (H.E.B Grocery Stores) *
Zip Code
Grocery Store
Location
Average
Household
Income (x)
Organic
Vegetables
Offered (y)
78204
78207
78204
78201
78212
78202
78237
78228
78227
78240
78230
78251
78238
78223
78221
78224
78220
78209
78216
78223
78218
78213
78227
78244
78231
78239
78217
78251
78250
78230
78247
78247
78251
78247
78248
78232
78249
South Flores
N. Rosillo st
Nogalitos st
Frederickburg rd
Olmos
New Braunfels
Castroville
Culebra rd
Marbach rd
Babcock rd
Wurzbach rd
W Loop 1604 N
Bandera rd
S.New Braunfels
SW Military
S Zarzamora
W.W. White rd
East basse rd
San pedro
S.E Military dr
Austiin hwy
West Avenue
Valley Hi dr
Foster dr
N.W Military
Montogomery
Perrinbeiter rd
FM 471 west
Guilbeau rd
De Zavala
Thousand oaks
O’Connor rd
Potranco rd
Bulverde rd
NW Loop 1604
18140 San Pedro
9238 Loop 1004
$71,186
$34,234
$71,186
$48,760
$78,096
$40,506
$38,166
$50,398
$49,437
$66,073
$86,566
$78,176
$59,154
$50,252
$48,364
$56,274
$41,318
$125,145
$65,911
$50,252
$53,945
$59,072
$49,437
$72,080
$108,486
$70,530
$57,199
$78,176
$78,288
$86,566
$84,181
$84,181
$78,176
$84,181
$135,547
$92,946
$77,894
36
4
28
31
78
14
12
18
38
84
61
56
62
44
26
29
15
95
18
65
50
35
36
28
95
46
29
73
53
86
68
56
85
86
93
82
96
Using this data from San Antonio, TX, we will explore
whether there is a relationship between neighborhood
income and access to organic items at local grocery stores.
Explanatory (independent) variable: the variable that
predicts, explains, or influences a trend in the response
variable. This is the _______________.
Response (dependent) variable: the measured outcome.
Responds to trends in the explanatory variable. This is the
_________________.
In this example, which variable is the explanatory variable?
Why?
In this example, which variable is the response variable?
Why?
*Dataset compiled by student Linda Saucedo, Fall 2019
76
Describing scatterplots
Graphic inspired by mathisfun.com
Correlation: measures how two variables are _________________.
Positive correlations: as the x values increase, the y values also tend to ___________________.
Negative correlations: as the x values increase, the y values tend to ____________________.
Graphic inspired by mathisfun.com
Least Squares Regression Line (LSRL): a straight line that roughly puts half of your data ____________ it
and half ______________ it.
• More formal definition coming next lesson.
Strong correlations: data is __________ to the LSRL
• The LSRL is a __________________ for the data
• If you used the LSRL to predict new data, you would make ________________________.
Weak correlations: data is ______ from the LSRL
• The LSRL is a __________________ for the data
• If you used the LSRL to predict new data, you would may be _________________________.
Direction:
Strength:
77
C – Context
D – Direction (positive/negative)
O – Outliers
F – Form (linear/non-linear)
S – Strength (strong/moderate/weak)
Put it all together: Describe the relationship between average
household income in a zip code and the number of organic
vegetables offered at the local grocery store…
Correlation coefficient (r)
Graphic inspired by mathisfun.com
𝑟=1
𝑟 = 0.91
𝑟 = 0.48
𝑟=0
𝑟 = −0.48
𝑟 = −0.91
𝑟 = −1
Correlation Coefficient (r): A number between _______________ that tells you the strength and
direction of a correlation.
Strength:
r close to 0 → _________ correlation
r close to -1, 1 → __________ correlation
Direction:
Negative r value → ______________ correlation
Positive r value → ______________ correlation
78
Lesson 3.1 Discussion
From 1994-2020, a statistician collected three pieces of data each summer at a beach:
1. The average temperature
2. The amount of ice cream sold at the beach shop
3. The amount of drownings
A
A) Discussion Question: Describe the
relationship between temperature and ice
cream sales. Does this relationship make sense?
Why or why not?
B
B) Discussion Question: Describe the
relationship between ice cream sales and
drownings. Does this relationship make sense?
Why or why not?
79
Does ice cream really cause drownings? No.
What’s really going on here? There is an ________________ third variable, temperature, that causes ice
cream sales and drownings to _____________.
Ice cream sales and drownings are correlated but do ________ cause one another.
Correlation ___ Causation
Sometimes, correlations…
1. Have an ___________________________ (3rd variable)
2. Cause one another, but not in assumed _________________
3. Are completely ______________, with no apparent cause
Completely Random Correlations, courtesy of Tyler Vigen: https://www.tylervigen.com/spurious-correlations
80
3.1 PRACTICE!
1. The table below shows data for 13 students in a statistics class. Each member of the class ran a 40-yard sprint and
then did a long jump (with a running start).
Sprint Time (s)
5.41 5.05 9.49 8.09 7.01 7.17 6.83 6.73 8.01 5.68 5.78 6.31 6.04
Long Jump Distance (in) 171 184 48
151 90
65
94
78
71
130 173 143 141
A. Create and label a scatterplot of the data:
B. Describe and interpret the scatterplot above.
C. What is the correlation coefficient? What does it mean?
81
2. A student wonders if tall women tend to date taller men than do short women. She measures herself, her dormitory
roommate, and the women in the adjoining rooms. Then she measures the next man each woman dates. Here are the
data (heights in inches):
A. How would r change if all the men were 6 inches shorter than the heights given in the table? Does the correlation tell
us if women tend to date men taller than themselves?
B. If heights were measured in centimeters rather than inches, how would the correlation change? (There are 2.54
centimeters in an inch.)
3. Consider each of the following relationships:
A. the heights of fathers and the heights of their adult sons
B. the heights of husbands and the heights of their wives
C. the heights of women at age 4 and their heights at age 18.
Rank the correlations between these pairs of variables from highest to lowest. Explain your reasoning.
82
Lesson 3.1: Day 1: How many rubber bands does Barbie need?
How many rubber bands should we attach to Barbie so that she has the absolute most
fun without smashing her head if she were to jump from the top of the bleachers in the
gym? Here’s the catch: You many only use 7 rubber bands to figure this out.
Complete the table:
# Rubber
0
bands
Distance
traveled
1
2
3
4
5
6
Use your group’s data to complete the following:
1. Identify the explanatory and response variables.
2. How many variables do we have? Are they categorical or quantitative?
3. Make a scatterplot.
4. Describe the relationship displayed in the scatterplot.
83
7
Lesson 3.1 – Displaying Relationships: Scatterplots
Big Ideas:
Check Your Understanding:
1. Is there a relationship between the amount of sugar (in grams) and the number of
calories in movie-theater candy? Here are the data from a sample of 12 types of
candy.
a. Identify the explanatory and response variables. Explain your reasoning.
b. Make a scatterplot to display the relationship between amount of sugar and the
number of calories in movie-theater candy.
c. Describe the relationship shown in the scatterplot.
84
AP Statistics Handout: Lesson 3.2
Topics: least squares regression line, slope, y-intercept, predictions using LSRL, extrapolation
Lesson 3.2 Guided Notes
Least squares regression line (LSRL)
Closing the school achievement gap with attendance: Low-income students tend to have lower
attendance rates and lower math test scores than their middle/upper income peers. Would raising their
attendance close the achievement gap? To explore this possibility, a random sample was collected of
students in Texas. For each student, data was collected on their attendance rate (percent of school days
attended) and raw test scores on the Algebra 1 state exam.
STAAR
Percent
Algebra 1
Attendance
Raw Score
(x)
(y)
95
89
67
98
99
76
92
91
76
85
82
45
42
31
51
49
38
46
41
35
39
37
Variables:
Explanatory:
________________
Response:
________________
Scatterplot:
How do we find
the exact equation
of the LSRL?
85
How can we use the information provided by the
squared residuals to determine which model (A or B)
better fits the data?
A
B
Least Squares Regression Line (LSRL): a linear model that minimizes the sum of the _________________
between the data and the model.
• Also called “line of best fit”
The LSRL model for our
data is the line that
minimizes the sum of the
residuals squared.
Slope and y-intercept
y-value
______________ y-value
Algebra 1: Linear Equation
Stats: Linear Regression
𝑦 = 𝑚𝑥 + 𝑏
𝑦ො = 𝑎 + 𝑏𝑥
𝑦ො = −7.69 + 0.57𝑥
_______
Linear Regression
𝑦ො = 𝑎 + 𝑏𝑥
_____________
𝑦ො = −7.69 + 0.57𝑥
𝑦ො: Predicted test score
𝑥: Percent attendance
86
𝑦ො = −7.69 + 0.57𝑥
1) Interpret the slope value:
Stem: For every 1 unit increase in explanatory variable, our model predicts an average
increase/decrease of slope in response variable.
Your answer:
2) Interpret the y-intercept:
Stem: When the explanatory variable is zero units, our model predicts that the response variable would
be y-intercept.
Your answer:
3) Is the y-intercept meaningful in this context? Explain…
Predictions using the LSRL
1) The superintendent of the district asks: “If a student meets our minimum attendance goal (87%),
what would their predicted test score be?” Answer his question and show your work (including drawing
your process on the scatterplot):
𝑦ො = −7.69 + 0.57𝑥
87
Lesson 3.2 Discussion
Dangers of prediction
In the past several years, superintendents have piloted large-scale (and sometimes quite expensive)
initiatives to improve student attendance. These included:
• Call programs for chronically absent students
• Hiring attendance case managers and coordinators
• Using Uber/Lyft for students with transportation issues
The result:
Attendance
Test Scores
Discussion Question: Why didn’t test scores grow when attendance rose?
Attendance research: Pyne, Grodsky, et al., (2018). What Happens When Children Miss School? Unpacking
Elementary School Absences in MMSD. Madison, WI: Madison Education Partnership.
88
Lesson 3.2 Practice
1) Dr. Youfa Wang at University of North Carolina published a study* on obesity in America. Using linear
regression, the study concluded that by 2048, if trends continue, 100% of Americans would be
overweight. Using the graphs** below, do you believe this conclusion is correct? Why or why not?
Example inspired by Ellenberg, J. How Not to be Wrong, pg. 50-61.
Data and Model
Model Projection
*Wang, Beydoun, et al., “Will all Americans become overweight or obese? estimating the progression and cost of the US obesity epidemic.”
Obesity (Silver Spring). 2008;16(10):2323‐2330. doi:10.1038/oby.2008.351
**Graphs provided are representative approximations of analyses from the paper
89
3.2 PRACTICE!
1. The following data shows the number of miles driven and advertised price for 11 used Honda CR-Vs from the 20052009 model years (prices found at www.carmax.com). The scatterplot below shows a strong, negative linear association
between number of miles and advertised cost. The line on the plot is the regression line for predicting advertised price
based on number of miles.
Thousand
Miles Driven
Cost
(dollars)
22
29
35
39
45
49
55
56
69
70
86
17998
16450
14998
13998
14599
14988
13599
14599
11998
14450
10998
A. Calculate the correlation. What does this value mean in plain English? What is the relative strength of the association?
B. What is the least squares regression equation for this association? Define any variables used.
C. Determine the y-intercept of the regression equation and interpret the value in context. Does the value have any realworld implications?
D. Determine the slope of the regression equation and interpret the value in context. Does the value have any realworld implications?
90
2. For a project, two AP Statistics students decided to investigate the effect of sugar on the life of cut flowers. They
went to the local grocery store and randomly selected 12 carnations. All the carnations seemed equally healthy when
they were selected. When they got home, the students prepared 12 identical vases with exactly the same amount of
water in each vase. They put 1 tablespoon of sugar in three vases, 2 tablespoons of sugar in three vases, and 3
tablespoons of sugar in three vases. In the remaining 3 vases, they put no sugar. After the vases were prepared and
placed in the same location, the students randomly assigned one flower to each vase and observed how many hours
each flower continued to look fresh. A scatterplot, residual plot, and computer output from the regression are shown.
Only 10 points appear on the scatterplot and residual plot since there were two observations at (1, 204) and two
observations at (2, 210).
Predictor
Constant
Sugar
Coef
181.200
15.200
S = 7.52596
SE Coef
3.635
1.943
R-Sq = 86.0%
T
49.84
7.82
P
0.000
0.000
R-Sq(adj) = 84.5%
A. What is the equation of the least-squares line? Be sure to define any variables you use.
B. Is a line an appropriate model for these data? Justify your answer. (You need at least two sentences.)
C. Interpret the value of “s” in the context of this problem.
D. Interpret the value of r2 in the context of this problem.
91
How safe is Barbie?
How can we be sure that the bungee cord we make Barbie will keep her safe?
Should you be worried if you used the wrong units, chose the wrong axes, or
measured wrong?
Below is the data for one group’s Barbie bungee.
Number of rubber bands
0
1
2
3
4
5
6
7
Distance traveled (in)
12
14
17
18
22
23
26
30
1. Go to stapplet.com and
create a scatterplot. Sketch
it to the right. (Leave extra
room on the right.)
2. Find the correlation. r = ______
3. How safe do you feel Barbie’s bungee
jump would be if we use this data? Use
the correlation to justify.
4. One of the group members snuck some extra rubber bands to collect more data.
Add the point (15 rubber bands, 49 in) to your scatterplot. How do you think this
outlier will affect the correlation? Verify in the applet. What is the new r?
5. One group member accidently left off a digit. Add the point (6 rubber bands, 6 in)
to the scatterplot. How do you think this outlier will affect the correlation? Verify
in the applet. What is the new r?
Unfortunately, the group had measured the lowest point of Barbie’s head in inches
instead of centimeters. To fix this they multiplied the inches by 2.54 (1 in. = 2.54 cm).
The new data is below.
Number of rubber bands
0
1
2
3
4
5
6
7
Distance traveled (cm)
30.48
35.56
43.18
45.72
55.88
58.42
66.04
76.20
6. How do you think these changes will affect the correlation? Verify by calculating
the correlation in the applet.
92
Displaying Relationships: Correlation
Important Ideas:
Check Your Understanding:
Fueleconomy.gov gives the city and highway fuel economy for all makes and models of
vehicles back to 1984. The scatterplot displays the city and highway fuel economy (mpg) for a
random sample of ten 2021 vehicles.
City fuel economy
(mpg)
Highway fuel
economy (mpg)
14.4
24.3
27.2
29.9
20.4
28.8
20.9
23.2
28.6
25.4
25.5
37.4
36.5
45.5
28.7
46.1
33.6
38.3
41.3
35.3
a. The correlation between city fuel economy and highway fuel economy for these 10 vehicles
is r = 0.917. Interpret this value.
b. If fuel economy was measured in feet per
gallon, rather than miles per gallon how would
the value of the correlation be affected?
Explain.
c. The Rolls-Royce Ghost EWB gets 14.4 city mpg and 25.5 highway mpg. What affect does
this point have on the correlation? Explain.
93
AP Statistics Handout: Lesson 3.3
Topics: residuals, residual plots, s, r2, outliers
Lesson 3.3 Guided Notes
Residuals
Residuals: The error (vertical distance) between a linear model’s _______________ and the __________
data point.
Residual = observed y – predicted y
̂
Residual = 𝒚 − 𝒚
To explore the relationship between attendance and test scores, a random sample of students’
attendance at school and their math test scores was collected. A linear regression was performed,
resulting in the linear equation: 𝑦̂ = −7.69 + 0.57𝑥, where 𝑦̂ = predicted raw score and x = percent of
school days attended.
a) Prediction: A new student comes to the
school. If his attendance rate is 80%, what
is his predicted test score?
𝑦̂ = −7.69 + 0.57𝑥
b) Error: The student gets 44 questions
correct on his exam. Find and interpret our
model’s prediction error (residual) for this
student.
Stems for interpreting residuals:
• 𝑦 − 𝑦̂ > 0 (positive residual): The actual response variable is greater than predicted by residual
units.
• 𝑦 − 𝑦̂ < 0 (negative residual): The actual response variable is less than predicted by residual units.
• Example: The actual pulse rate is greater than predicted by 3.65 beats per minute.
94
Residual Plots
Residual Plots – What a “Good Fit” Looks Like…
Residuals ______________ scattered around 0, without
a pattern → linear model is a good fit for this data!
Residual Plots – What a Non-Linear Fit Looks Like…
Residuals show a _____________
pattern → linear model is ______
a good fit for this data!
Residual Plots – Heteroskedastic
More _____________ in residuals
as x values increase → linear model
is ______ a good fit for this data!
95
Standard Deviation of the Residuals (s)
Standard deviation: Typical distance between data points and their mean
Standard deviation of the residuals (s): ______________ error between data points and their LSRL
(typical _______________)
s: The “typical” residual length
s:
Stronger correlation
• _________________
s:
Weaker correlation
• _________________
1) The standard deviation of the residuals for the LSRL between attendance and test scores is s = 1.99.
Interpret this value.
Stem for interpreting s:
• When using the LSRL with explanatory variable to predict response variable, we will typically be off
by about value of s with units of the response variable (y).
96
The Data – Mass Shootings
Which variable most strongly predicted the annual frequency of U.S. mass shootings? Which model had
the smallest residuals?
Year
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
U.S.
Public
Mass
Shootings
1
1
1
2
3
5
1
1
0
1
1
2
3
4
3
4
1
3
7
5
4
7
6
11
12
10
Rifles
manufactured
in U.S.
1,316,607
1,411,120
1,424,315
1,251,341
1,535,690
1,569,685
1,583,042
1,284,554
1,515,286
1,430,324
1,325,138
1,431,372
1,496,505
1,610,923
1,734,536
2,248,851
1,830,556
2,318,088
3,168,206
3,979,570
3,379,549
3,691,799
4,239,335
2,504,092
U.S.
residents
w/ mental
health
disorders
38,705,887
39,220,805
39,949,663
40,703,543
41,470,048
42,210,935
42,979,125
43,446,520
43,873,630
44,268,083
44,676,753
45,100,679
45,380,842
45,643,428
45,887,405
46,131,343
46,360,267
46,664,296
46,955,985
47,208,307
47,445,735
47,775,445
48,106,294
Revenue
from U.S.
Video Game
Sales (in
billions of
2020 dollars)
14.705
13.272
11.41
12
13.973
16.94
16.39
17.255
17.732
16.541
14.96
15.065
17.145
23.436
26.18
23.64
21.948
18.924
19.152
19.14
24.192
25.38
26.215
30.555
36.516
37.9
Notes about the data:
•
Public mass shootings are defined as having 4+ victims,
indiscriminate targeting. Excludes shootings in private
residences or stemming from other crimes such as robbery.
•
Mass shooting data from Washington Post and Mother Jones: https://www.washingtonpost.com/graphics/2018/national/mass-shootings-in-america/
•
Rifle data from Bureau of Alcohol, Tobacco, Firearms and Explosives 2019 Statistical Update: https://www.atf.gov/resource-center/data-statistics
•
Mental health data from Institute for Health Metrics and Evaluation via Our World in Data: https://ourworldindata.org/mental-health
•
Video game data from The NPD Group: https://www.npd.com/wps/portal/npd/us/industry-expertise/video-games/
Rifle production as explanatory variable:
s = 1.31
Mental health patients as explanatory variable:
s = 1.69
Compared to mental health, rifle production is the __________________ predictor of mass shooting
frequency
97
The coefficient of determination (r2)
Graphic inspired by mathisfun.com
𝑟=1
𝑟 = 0.91
𝑟 = 0.48
𝑟2 = 1
𝑟 2 = _____
𝑟 2 = _____
𝑟=0
𝑟 = −0.48
𝑟 2 = _____
Squaring the values:
1) Gets rid of _____________
2) Emphasizes differences in ______________
𝑟 2 = _____
𝑟 = −1
𝑟 = −0.91
𝑟 2 = _____
𝑟 2 = _____
_____ ≤ 𝑟 2 ≤ _____
𝑟 2 close to 0 → __________ correlation
𝑟 2 close to 1 → ______________ correlation
Stem for interpreting r2 :
𝑟 2 % of the variation in response variable can be explained by the linear relationship with explanatory
variable
100%. The linear model _____________
explains the data’s pattern.
𝑟 2 = 1.00 = 100%
72%. The linear model explains _________
of the data’s pattern, but not all of it.
𝑟 2 = 0.72 = 72%
1) The linear relationship between U.S. rifle
production and the annual frequency of mass
shootings has an 𝑟 2 value of 0.62. Interpret
this value.
98
r2 = 0.36
r2 = 0.62
r2 = 0.69
Video game sales were the _______________
_______________ of the annual frequency of
public mass shootings in the United States.
The relationship between s and r2
Strong:
• Low s
• High r2
Weak:
• High s
• Low r2
The effect of outliers
Which of the following models have an outlier? Explain your reasoning.
A
C
N
B
N
99
2017 was an outlier: A data value
_______________ far away from the
_____ (unusually large residual)
Without Outlier
With Outlier
r = 0.79
r2 = 0.62
s = 1.31
r = _____
r2 = _____
s = _____
r, r2, and s are _____ resistant to outliers. Outliers ___________ the strength of a relationship, making…
o r ____________ (magnitude)
o r2 ___________
o s ___________
Lesson 3.3 Discussion
Video game sales were the strongest predictor of the annual frequency of public mass shootings in the
United States. Almost 70% of the variation in yearly public mass shootings can be explained by the linear
relationship with video game sales (r2 = 0.69).
Discussion Question: Would banning violent
video games prevent more mass shootings
than banning assault weapons or improving
mental health services? Why or why not?
100
Remind you of anything?
101
AP Statistics – AP Exam Review
Linear Regression & Computer Output: Interpreting Important Variables
II. More Practice with Linear Regression and Residual Plots
Fast food is often considered unhealthy because much fast food is high in fat and calories. The fat and calorie content for
a sample of 5 fast-food burgers is provided below.
Fat(g)
Calories
31
580
35
590
39
640
39
680
43
660
a) Identify the explanatory and the response variables:
b) Use the calculator to make a scatter plot of these ordered pairs. Sketch the scatter plot here.
c) What information does the scatter plot provide? That is, use the scatter plot to describe the
relationship between the fat grams and calories in a fast food burger.
d) Find the following summary statistics for this data:
x, y, sx , s y
e) Now use your calculator to record the following statistics and to find the equation of the least squares
line. Record the equation and use it for the remaining computations.
a, b, r 2 , r,
102
AP Statistics – AP Exam Review
Linear Regression & Computer Output: Interpreting Important Variables
f)
Examine a graph of the least squares line superimposed on your scatter plot.
Stat > calc > 8:linreg(a+bx) >L1, L2, Y
To get the Y to show up: Vars > Y Vars> 1:Function > 1: Y1
This will graph the LSRL along with your scatterplot. If you go to the Y= screen, you will now see
the equation for the LSRL
g) Does the line appear to be good model for the data?
h) What is the value of your slope? What information does it provide? Be specific.
i)
How many calories would you predict a burger with 20 fat grams has?
j)
Calculate the residual for 35 fat grams.
k) Calculate the value of r 2 . What information does it provide? Be specific.
l)
What is the value of r ? What does it tell you in this situation?
m) Make a residual plot on your calculator. Be sure to label both axes with words and a “friendly” scale.
n) Based on this residual plot, do you think the least squares line is a good model for this data
103
AP Statistics – AP Exam Review
Linear Regression & Computer Output: Interpreting Important Variables
Becky’s parents have kept records of her height since she was born. The data set consists of Becky’s age in
months and her height in centimeters. The summary statistics for the data are provided below:
std. dev. age: 8.5 months
Mean age: 44 months
Mean ht: 82 cm
std. dev. ht: 4.1 cm
The correlation between age and height is .86.
(a) Find the equation of the least squares line that you would use to predict Becky’s height from her age. Show
all work.
(b) What real-world information does the slope provide? Be specific!
(c) Suppose height had been measured in inches rather than in centimeters. What would be the
correlation between age and height in inches? Note: 1 inch = 2.54 cm
104
How many iPhones will be sold?
Here is the data of all iPhone sales during their opening weekends:
iPhone
Year
Original
3G
3Gs
4
4S
5
5C, 5S
6, 6 Plus
6S, 6S Plus
2007
2008
2009
2010
2011
2012
2013
2014
2015
Units Sold
(millions)
0.5
1
1
1.7
4
5
9
10
13
1. Use stapplet.com to create a scatterplot of the data with year as the explanatory
variable and units sold as the response. Sketch the scatterplot in the space above.
2. Describe the form of the distribution.
3. Use the applet to find the least squares regression line. Write the equation below and
graph it on your scatterplot above.
4. Use the least squares regression line to calculate the residual for 2007. Interpret the
residual.
5. Complete the table below.
Year
2007
2008
2009
2010
2011
2012
2013
2014
2015
Actual
Units Sold
(millions)
0.5
1
1
1.7
4
5
9
10
13
Predicted
Units Sold
(millions)
Residual
6. Graph the residuals on the axes
below. This is called a residual plot.
7. For which points was the actual greater than the predicted? Which were less than
predicted? Identify these on the graph.
8. Do you think the regression line is a good fit for the data? Why or why not? Explain using
the residual plot.
105
LSRL and Residual Plots
Important Ideas:
Check Your Understanding:
Fueleconomy.gov gives the city and highway fuel economy for all makes and models of vehicles
back to 1984. The table gives the city and highway fuel economy (mpg) for a random sample of
ten 2021 vehicles.
City fuel economy
(mpg)
Highway fuel
economy (mpg)
14.4
24.3
27.2
29.9
20.4
28.8
20.9
23.2
28.6
25.4
25.5
37.4
36.5
45.5
28.7
46.1
33.6
38.3
41.3
35.3
a. Calculate the equation of the least-squares regression line.
b. Make a residual plot for the linear model in Question 1.
c. What does the residual plot indicate about the appropriateness of the linear model? Explain
your answer.
106
AP Statistics Handout: Lesson 3.4
Topics: leverage and influential points, regression from calculators and computers
Lesson 3.4 Guided Notes
Leverage and Influential Points
The LSRL goes through the point: __________
̅ = $𝟔𝟗, 𝟏𝟗𝟖
𝒙
̅ = 𝟓𝟐 items
𝒚
Dataset compiled by student Linda Saucedo, Fall 2019
Removing low leverage points
Removing high leverage points
Low leverage points are close to ________
Low leverage points ___________ affect LSRL much
High leverage points are far from ________
High leverage points have a _________ effect
on an LSRL’s slope.
Influential points: points that, if removed, change the slope, y-intercept, or correlation substantially.
• Three types: Outliers (change ____________), High-Leverage (change _______________), Both
High-Leverage
Outlier
107
Both
Using your Calculator for Regression
1. Turn ON Stat Diagnostics
(MODE)
2. Input data into L1, L2
(STAT → Edit…)
3. Find LinReg
(STAT → Calc → 8:LinReg)
4. Click “Calculate”
Reading Computer Regression Tables
Identify the equation of the LSRL and the important elements from this computer regression output:
Predictor
Coef
SE Coef
T
P
Constant
-14.7
9.30
-1.58
0.122
Income
0.001
0.0001
7.51
0.000
S = 17.46
R-Sq = 61.7%
R-Sq (adj) = 60.6%
108
𝑦̂ = 𝑎 + 𝑏𝑥
Prediction as Discrimination?
1) The grocery store chain (H-E-B) wants to open two new locations in San Antonio - one in a lower
income zip code (mean income: $35k) and one in a higher income zip code (mean income: $115,000).
Based on our LSRL, how many more organic items are predicted to be offered at the location in the
higher income neighborhood?
Lesson 3.4 Discussion
Discussion Question: An investigative journalist finds that a different supermarket (not H-E-B) actually
uses a model to estimate how many organic items it should put on shelves purely based on
neighborhood income. The model calls for fewer organic items in low-income areas.
• Name one reason this practice may be unethical.
• If you were the company’s CEO, how would you justify using such a model?
• In your view, is this an unethical use of data? Why or why not?
109
AP Statistics – AP Exam Review
Linear Regression & Computer Output: Interpreting Important Variables
I. Minitab / Computer Printouts
Below is a computer output. You will be expected to use and interpret computer output on the
AP Exam. This output is from Minitab, however most computer output looks very similar. We will
discuss which numbers you need to know, what they mean, and how to interpret them.
SE Coef, = 0.3839 represents the standard deviation of the slope
S = 13.40 represents standard deviation of residuals
Constant ----- -87.12
This is the y-intercept. This is the value of the response variable when the explanatory variable is 0.
Check the context of the situation. Often, there can be no such value.
In this case, it is not possible to have a volume that is negative nor is it possible to have a height of zero.
Height/Slope --- 1.5433
This is the coefficient of the explanatory variable, thus it is the slope.
This entire line of numbers deals with regression for slope.
For each increase in height of one unit, the volume is expected to increase by approximately 1.5433
units. (Actual units were not provided)
Prediction equation----- y
87.1 1.54x (ie. Least Squares Regression Line)
This is an equation used to make predictions and is based on only one sample.
SE Coef --------- 0.3839
This is the standard deviation of the slope. Remember, this data came from only one sample. We would
expect the slope to vary a little from sample to sample. Thus,
If we gathered repeated samples, we would expect the slope of the volumes of the trees to vary
by approximately 0.3839 units.
S --------- 13.40
This is the standard deviation of the residuals. The average amount that the observed values differ from
the predicted values is 13.40. The average amount that the observed volumes of trees differ from the
predicted volumes is approximately 13.40 units.
110
AP Statistics – AP Exam Review
Linear Regression & Computer Output: Interpreting Important Variables
r 2 ------------- 35.8%
This is the correlation of determination, which is the fraction or proportion of variation in the y values
that is explained by the least squares regression of y on x.
About 35.8% in the variation in volume can be explained by the least squares regression of y( volume)
on x ( height).
r
r2
.358 .598
This is the correlation coefficient. It tells you strength and direction of the relationship.
With an r value of .598, there is a weak, positive relationship between height and volume of trees.
T ------------ 4.02
This is the test statistic which = test statistic
statistic parameter
std .dev of statistic
b
SE
P--------- 0
This is the p-value of a Linear Regression t test.
With a p-value of approx.. 0 less than any alpha level (.05, .01), reject the null. There is evidence that
there is a relationship between the volume of a tree and its height.
Example 2: Minitab / Computer Printouts
Regression Analysis: Height versus Mother Height
The regression equation is
Height = 24.7 + 0.640 Mother Height
(dependent
variable, y)
(intercept,
b0 )
(slope,
b 1)
(estimates)
Predictor
Constant
Mother H
Coef
24.690
0.6405
the estimated regression
equation: ŷ = b0 + b1x
(independent
variable, x)
(sd of ests.)
(test
statistics)
SE Coef
8.978
0.1394
T
2.75
4.59
intercept, b0
slope, b1
S = 2.973
(standard error
of estimate, se)
R-Sq = 35.7%
(coefficient of linear
determination, r2)
(p-values)
P
0.009
0.000
IGNORE these values
tests H0: β1 = 0, vs. two-tailed altern.
(the latter is equivalent to testing for
linear correlation between x and y)
R-Sq(adj) = 34.0%
(adjusted r 2, used for multiple regression)
* (the coefficient of linear correlation is the square root
of r2, with the same sign as the slope, b1)
for a simple linear regression
minitab models only conduct
two-tailed alternatives
S – 2.973 standard error of the estimate, or the standard deviation of the residuals - actual values
versus the predicted values
SE Coef – 0.1394 – standard deviation of the slope
111
AP Statistics – AP Exam Review
Linear Regression & Computer Output: Interpreting Important Variables
Practice Minitab / Computer Printouts
1. A sample of men agreed to participate in a study to determine the relationship between several variables
including height, weight, waste size, and percent body fat. A scatterplot with percent body fat on the y-axis and
waist size (in inches) on the horizontal axis revealed a positive linear association between these variables.
Computer output for the regression analysis is given below:
Dependent variable is: %BF
R-squared = 67.8%
S = 4.713 with 250-2 = 248 degrees of freedom
Variable
Constant
Waist
Coefficient
-42.734
1.70
se of coeff
2.717
0.0743
t-ratio
-15.7
22.9
prob
<.0001
<.0001
(a) Write the equation of the regression line (be sure to use correct notation and define your variables):
(b) Explain/interpret the information provided by R-squared in the context of this problem. Be specific.
(c) Calculate and interpret the correlation coefficient (r).
(d) One of the men who participated in the study had waist size 35 inches and 10% body fat. Calculate the
residual associated with the point for this individual.
112
AP Statistics – AP Exam Review
Linear Regression & Computer Output: Interpreting Important Variables
Determine the LSRL, Standard deviation for slope, correlation coefficient and the standard error of
the residuals for each:
#2.
#3.
113
How good are the predictions for Barbie?
Here is the data from one of the groups. The group forgot to record their measurement
for 5 rubber bands.
Number of rubber bands
0
1
2
3
4
5
6
7
Distance traveled (cm)
25
32
41
49
55
?
69
78
1. Go to stapplet.com to make a scatterplot. Then click “Calculate least-squares
regression line”. This is the line that best models the data. Write the equation below.
2. Use the regression line to predict the distance Barbie travels for 5 rubber bands.
Show work.
3. One of the group members later found the measurement for 5 rubber bands was 64
cm. Was the prediction from #2 too high or too low? How far off?
4. Predict the distance that Barbie would travel if the group used 20 rubber bands.
Would you trust this prediction more or less than the prediction you made in #2?
5. What is the y-intercept of the equation of the regression line? What does it mean?
6. What is the slope of the equation of the regression line? What does it mean?
114
Prediction, Residuals, Interpreting a Regression Line
Important Ideas:
Check Your Understanding:
Michael is a runner. He uses his Apple watch to keep track of his distance and the number of
calories he burns for 20 runs. A scatterplot of y = calories burned and x = distance (in miles)
shows a fairly strong, positive linear relationship. The regression equation 𝑦" = 20 + 160𝑥
models the data fairly well.
a. Interpret the slope of the regression line.
b. Does the value of the y intercept have meaning in this context? If so, interpret the y
intercept. If not, explain why.
c. Predict the number of calories Michael burns if he runs 5 miles.
d. Calculate and interpret the residual if his Apple watch said that he burned 910 calories
on a 5-mile run.
e. Michael is thinking about signing up for his first marathon. So far, his longest run has
only been 10 miles. Should he use the regression equation to predict how many
calories he would burn if he runs a marathon (26.2 miles)? Explain.
115
How many iPhones will be sold?
Here is the data of all iPhone sales during their opening weekends:
iPhone
Original
3G
3Gs
4
4S
5
5C, 5S
6, 6 Plus
6S, 6S Plus
Year
(after
2000)
7
8
9
10
11
12
13
14
15
Units Sold
(millions)
0.5
1
1
1.7
4
5
9
10
13
1. Use stapplet.com to create a scatterplot of the data with year as the explanatory
variable and units sold as the response. Sketch the scatterplot in the space above.
2. Would you use a linear regression to model the data? Sketch the residual plot below to
support your explanation.
3. Since we expect that the data is nonlinear, we cannot make a linear regression.
However, we can transform the data to make it more linear. First we need to decide
what type of function we think the data would best fit so that we can transform it.
a. What type of model do you think best fits the data?
b. What is the general form of this model?
c. Algebraically, what is the inverse of that function?
d. How can we transform our data using this inverse?
116
4. Complete the table below.
Year
7
8
9
10
11
12
13
14
15
Actual
Units Sold
(millions)
0.5
1
1
1.7
4
5
9
10
13
log(Units
Sold
(millions)
5. Use stapplet.com to create a scatterplot of the data with year as the explanatory
variable and log(units sold) as the response. Sketch the scatterplot in the space above.
6. Calculate the LRSL for the transformed data and write it below.
7. Do you think the regression line is a good fit for the transformed data? Why or why not?
Explain using the residual plot and sketch it below.
8. According to this model, how many iPhones should be sold in 2015?
9. Calculate and interpret the residual for the actual number of iPhones sold in 2015.
117
Transforming Nonlinear Data
Important Ideas:
Check Your Understanding:
A party company specializes in creating balloon boxes, which are
boxes that are completely filled with small balloons. The number of
balloons needed to create a balloon box is a function of the box
height. Here is a scatterplot showing the number of balloons needed
to fill boxes of various heights. Note the clear curved form.
Because height is one-dimensional and number of balloons to fill the
box (volume) is three-dimensional, a power model of the form
number of balloons = a(height)3, should describe the relationship.
Here is a scatterplot of number of balloons versus height3. Because
the transformation made the association roughly linear, we used
computer software to perform a linear regression analysis of y =
number of balloons versus x = height3.
Regression Analysis
Predictor
Coef
SE Coef
Constant
45.008
29.87
Height^3
10.809
0.197
S = 76.938
R-Sq = 99.6%
T
P
1.507
0.15
54.818
0.000
R-Sq(adj) = 99.5%
a. Give the equation of the least-squares regression line. Define any variables you use.
b. Suppose you want to fill a box with a height of 8 feet. Use the model from part (a) to predict
the number of balloons that would be needed.
118
Can You Guess My IQ?
As part of a new transcript at our school, the counselors have decided to include an IQ
score in addition to GPA. Will knowing the GPA help to predict IQ?
Five students requested that the counselors update their transcripts for them: Adam, Bernard,
Christie, Deja, and Eldin. Their IQ scores are 110, 85, 120, 95, 105 but they have been all
mixed up and the counselors don’t know which IQ score goes with which student.
The guidance counselors are forced to predict the IQ for each student. Each counselor takes
a different approach.
Counselor #1: The New Guy
The New Guy is so nervous about being wrong, so he wants to play it safe with his
predictions and minimize his error. He decides to find the average IQ and use it as his
prediction for all five of the students:
Student
Predicted IQ
Adam
Bernard
Christie
Deja
Eldin
Counselor #2: The Veteran
The Veteran noticed an equation written on the board in the AP Statistics room:
IQ = 16•GPA +57.3 . She realized that GPA can help her to make better predictions. She
looks up the GPA of each student:
Adam
Bernard
Christie
Deja
Eldin
GPA = 1.8
GPA = 2.4
GPA = 2.9
GPA = 3.4
GPA = 3.8
Then she used the line of best fit to make her predictions. IQ = 16•GPA +57.3
Student
GPA
Predicted IQ
Adam
1.8
Bernard
2.4
Christie
2.9
Deja
3.4
Eldin
3.8
Counselor #3: The Truth Seeker
Guidance counselor #3 pulled the five students out of class and found the truth.
Student
GPA
Actual IQ
Adam
1.8
85
Bernard
2.4
95
Christie
2.9
110
Deja
3.4
105
119
Eldin
3.8
120
Who made the better predictions?
Now let’s see which counselor made better predictions:
Counselor #1: The New Guy (used the mean IQ for every prediction)
Student
Adam
Bernard
Christie
Deja
85
95
110
105
Actual IQ
Predicted IQ
Error (Actual - Predicted)
Squared error
Eldin
120
Sum of the squared errors:
Counselor #2: The Veteran (used the line of best fit for every prediction)
Student
Adam
Bernard
Christie
Deja
Eldin
GPA
1.8
2.4
2.9
3.4
3.8
Actual IQ
85
95
110
105
120
Predicted IQ
Error (Actual - Predicted)
Squared error
Sum of the squared errors:
Who did better? Why?
Counselor #1 sum of squared errors: ______ Counselor #2 sum of squared errors: _______
Calculate the percentage improvement of the sum of squared errors from Guidance
Counselor #1 to Guidance Counselor #2.
We also could find each counselor’s typical error. Use algebra to turn the sum of the squared
errors into the “average” error for Guidance Counselor #2.
Find the correlation (r), the coefficient of determination (r2) and the standard deviation of the
residuals (s) for the data using the calculator.
GPA
1.8
2.4
2.9
3.4
3.8
Actual IQ
85
95
110
105
120
r=
r2 =
s=
120
s and r2
Important Ideas:
Check Your Understanding:
Fueleconomy.gov gives the city and highway fuel economy for all makes and models of vehicles
back to 1984. The table gives the city and highway fuel economy (mpg) for a random sample of ten
2021 vehicles.
City fuel economy (mpg)
Highway fuel economy
(mpg)
14.4 24.3 27.2 29.9 20.4 28.8 20.9 23.2 28.6 25.4
25.5 37.4 36.5 45.5 28.7 46.1 33.6 38.3 41.3 35.3
Analyze the data using the two-quantitative variable option on stapplet.com.
a. Find the equation of the LSRL. Write it below.
b. Find and interpret s.
c. Find and interpret the value of 𝑟 ! .
121
How do outliers affect the LSRL?
1. Use the Correlation and Regression applet at www.tinyurl.com/regressionapplet
• Click on the graphing area to add 10 points in the lower-left corner so that the
correlation is about r = 0.50.
• Check the boxes to show the LSRL and the mean X and Y lines.
• Sketch it below.
2. For each of the following situations add the point to the scatterplot and decide if the slope,
y-intercept and correlation will increase or decrease.
a. If a point is added on the far right side of the graph on the horizontal line for the
mean of Y.
Slope:
y-intercept:
Correlation:
b. If a point is added on the far left side of the graph on the horizontal line for the mean
of Y.
Slope:
y-intercept:
Correlation:
c. If a point is added below the LSRL on the vertical line for the mean of X.
Slope:
y-intercept:
Correlation:
d. If a point is added above the LSRL on the vertical line for the mean of X.
Slope:
y-intercept:
Correlation:
3. Which outliers had the greatest impact on the LSRL, vertical or horizontal outliers?
122
Outliers and the LSRL
Important Ideas:
Check Your Understanding:
You’ve probably heard the saying “Practice makes perfect!”, but does practice also help you
complete a task faster? A study was conducted to find out. A random sample of 15 high school
students were taught how to solve a Rubik’s cube. Then they were each randomly assigned a
number of times to practice this new skill. After they completed their assigned number of practices
they were timed solving the Rubik’s cube. Here is a scatterplot of the results along with the leastsquares regression line.
a. Describe the influence the student who was assigned
to practice following the steps to solve a Rubik’s cube
14 times has on the equation of the least-squares
regression line.
b. Describe the influence the student who was assigned to practice following the steps to solve a
Rubik’s cube 14 times has on the standard deviation of the residuals and r2.
c. The mean and standard deviation of the number of practices are 𝑥̅ = 8 practices and sx =
4.47 practices. The mean and standard deviation of time are 𝑦$ = 7.71 minutes and sy =
1.20 minutes. The correlation between number of practices and time to solve the Rubik’s
cube is r = –0.793. Find the equation of the least-squares regression line for predicting time
to solve the Rubik’s cube from the number of practices.
123
How close can you get to the finish line?
The Goal: Get your team’s car to reach the finish line without going over.
The Catch: The distance will not be revealed until later.
Test drive your car for pull-backs of 2, 4, 6, and 8 inches. Measure the distance the car
travels. Repeat this process 3 times. Fill in the table below.
x = Pull-back Distance (in.)
2
2
2
4
4
4
6
6
6
8
y = Distance Traveled (in.)
You will create three different regression models for the data and will decide which is
best. For each of the three models below:
a. Create a scatterplot
b. Calculate an LSRL
c. Analyze linearity using any strategies you choose. Show your work.
Linear: Plot (x, y):
Quadratic: Plot (x2, y):
Exponential: Plot (x, log y)
1. Which regression is the best choice for your data? Why?
2. Calculate how many inches you will have to pull back for a finish line of 80 inches.
124
8
8
Choosing the Best Regression
Important Ideas:
Check Your Understanding:
Gapminder.org is an organization dedicated to studying global health. Here is a scatterplot
showing average annual income (in thousands of U.S. dollars) and life expectancy (years) for a
random sample of 15 countries selected from the Gapminder data base.
The output shows three possible models for predicting life expectancy from income. Model (a) is
based on the original data, while Model (b) and (c) involve transformations of the original data.
Each set of output includes a scatterplot with a least-squares regression line added and a residual
plot. The regression equations are given below.
Model (a)
𝑙𝚤𝑓𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑎𝑛𝑐𝑦
= 69.9352 + 0.1530(𝑖𝑛𝑐𝑜𝑚𝑒)
Model (b)
Model (c)
ln (𝑙𝚤𝑓𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑎𝑛𝑐𝑦)
= 4.2436 + 0.0021(𝑖𝑛𝑐𝑜𝑚𝑒)
ln (𝑙𝚤𝑓𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑎𝑛𝑐𝑦)
= 4.1393 + 0.0603𝑙𝑛(𝑖𝑛𝑐𝑜𝑚𝑒)
1. Use each model to predict the life expectancy of residents of a country for which the average
annual income is $80,000.
2. Which model does the best job summarizing the relationship between income and life
expectancy? Explain your answer.
125
Chapter 3 Review Sheet
In my meeting yesterday, I noticed that the number of people paying attention to
the training was declining throughout the day. I collected the following data:
Time (hours)
# people paying attention
0
80
1
68
2
62
3
50
4
48
5
32
6
21
1. Make a scatterplot of the data on your calculator and sketch it below.
2. Describe the data
3. Find the line of best fit using the equations for a and b. Confirm that it
matched with the calculator’s line of best fit.
4. Make a prediction for the # of people paying attention at t = 2.5 hours.
5. What is r? Interpret.
6. What is r2? Interpret.
7. What is the slope of the LSRL? Interpret in the context of the problem.
8. What is the y-intercept of the LSRL? Interpret in the context of the problem.
126
9. Identify the values a and b in the Minitab output:
Predictor
Constant
Time
Coef
79.75
-9.393
SE Coeff
5.34
4.23
S = 3.143
R-Sq = 98.0%
T
3.41
-15.81
P
0
0
R-Sq (adj) = 97.2%
10. Interpret s = 3.143 in the context of the problem.
11. Find the residual for a time of 4 hours.
12. Make a residual plot on your calculator. Sketch it below.
13. Mr. Bakri walked into the meeting at t = 7 hours and all of a sudden 82
people were paying attention. Would this point be influential? Prove it.
127
Barbie Bungee – The Finale
It’s finally time to jump Barbie! At the end of the hour we will be dropping Barbie from
the staircase in the foyer which is 17 ft. (5.2 m). Before we drop her, we will use
everything we’ve learned this chapter to calculate the best possible length of bungee
cord.
Write in your group’s data in the table below.
Number of rubber bands
0
1
2
3
4
5
6
7
Lowest point head reaches (cm)
1. Identify which variable is the explanatory variable and which is the response
variable?
2. Use the Applet to create a scatterplot.
3. Describe your distribution (DUFS).
4. Estimate the r value of your distribution.
5. What would happen to the correlation (r) if you graphed the scatterplot with the
lowest point on the horizontal axis and # rubber bands on the vertical axis?
128
6. Calculate the correlation using SPA applets. Write it below. What is the unit of
the correlation?
7. Use the Applet to find the least squares regression line for your data. Write the
equation below.
8. What is the slope of your LSRL? Interpret the slope.
9. What is the y-intercept of your line? Interpret.
10. Use the LSRL to calculate and interpret the residual for 4 rubber bands.
11. Sketch the residual plot for your LSRL.
12. Find the r2 value and interpret it.
13. Find the standard deviation of the residuals and interpret it.
14. Is the linear regression an appropriate model? Explain.
129
15. Transform the data to y vs. x2 to create a quadratic model. Write the equation of
the LSRL of the transformed variables below.
16. Calculate and interpret the residual using the quadratic regression for 4 rubber
bands.
17. Sketch the residual plot and record the r2 value for the quadratic regression.
r2 = __________
1. Transform the data to log y vs. x to create a exponential model. Write the
equation of the LSRL of the transformed variables below.
2. Sketch the residual plot and record the r2 value for the exponential regression.
r2 = __________
3. Which model is best?
4. Use your model to predict the number of rubber bands Barbie will need in order
to have the most exciting yet safe bungee jump from 17 ft. (518 cm)
130
2017 AP® STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
Part A
Questions 1-5
Spend about 65 minutes on this part of the exam.
Percent of Section II score—75
Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
1. Researchers studying a pack of gray wolves in North America collected data on the length x, in meters, from
nose to tip of tail, and the weight y, in kilograms, of the wolves. A scatterplot of weight versus length revealed
a relationship between the two variables described as positive, linear, and strong.
(a) For the situation described above, explain what is meant by each of the following words.
(i) Positive:
(ii) Linear:
(iii) Strong:
The data collected from the wolves were used to create the least-squares equation yˆ = -16.46 + 35.02 x.
(b) Interpret the meaning of the slope of the least-squares regression line in context.
(c) One wolf in the pack with a length of 1.4 meters had a residual of -9.67 kilograms. What was the weight
of the wolf?
131
STATISTICS
SECTION II
Part B
Question 6
Spend about 25 minutes on this part of the exam.
Percent of Section II score—25
Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
6. A newspaper in Germany reported that the more semesters needed to complete an academic program at the
university, the greater the starting salary in the first year of a job. The report was based on a study that used a
random sample of 24 people who had recently completed an academic program. Information was collected on
the number of semesters each person in the sample needed to complete the program and the starting salary, in
thousands of euros, for the first year of a job. The data are shown in the scatterplot below.
(a) Does the scatterplot support the newspaper report about number of semesters and starting salary? Justify
your answer.
The table below shows computer output from a linear regression analysis on the data.
Predictor
Constant
Semesters
S = 7.37702
Coef
34.018
1.1594
SE Coef
4.455
0.3482
R-Sq = 33.5%
T
7.64
3.33
P
0.000
0.003
R-Sq(adj) = 30.5%
(b) Identify the slope of the least-squares regression line, and interpret the slope in context.
132
An independent researcher received the data from the newspaper and conducted a new analysis by separating the
data into three groups based on the major of each person. A revised scatterplot identifying the major of each
person is shown below.
(c) Based on the people in the sample, describe the association between starting salary and number of semesters
for the business majors.
(d) Based on the people in the sample, compare the median starting salaries for the three majors.
(e) Based on the analysis conducted by the independent researcher, how could the newspaper report be modified
to give a better description of the relationship between the number of semesters and the starting salary for
the people in the sample?
133
134
Unit 3
Chapter 4
Collecting Data
135
AP Statistics Handout: Lesson 4.1
Topics: sample vs. population, biased sampling, simple random samples (SRS)
Lesson 4.1 Guided Notes
Sample vs. Population
Census: when you collect data on every individual in the _______________________.
Sample: a _____________ of individuals from a population.
Population
Sample
It’s hard to measure everyone (population), so we take a sample
and make ____________________ about the population.
Graphic courtesy of the National Center for Education Statistics
Biased Sampling
Example: Investigating College Advertisement Statistics
SW Tennessee Community College: The homepage of their website boasts: “Our overall graduation
placement rate is ______, with 91% working in their field of study.” - www.southwest.tn.edu (6/9/2020)
Population
Sample → _________
Among full-time, first-time degree or certificate-seeking students
who entered in 2010/2011, Source: IPEDS (2020)
Bias: a study flaw that leads to _______________________ and/or inaccurate estimates.
Undercoverage: When part of the population has a __________________ of being included in a sample.
-Leads to bias.
-Ex: excluding the students who didn’t graduate.
136
Rogers State University (Oklahoma): In a recent report*, the University found that about 75% of
graduates were pursuing another degree or had found full-time employment by their final semester.
The same report shows that the response rate to the University’s questions/surveys was only 20%.
Nonresponse: When individuals chosen for a sample don’t respond.
-Leads to bias if these individuals ___________ from respondents.
Population
Sample → _______________
Grads without jobs probably
respond _______________.
*Employment and Continuing Education for Graduating Students 2017-2019 AY 3-Year Aggregation
(downloaded 6/9/2020 from https://www.rsu.edu/about/accountability-academics/student-outcomes/)
Question: How could bias in the sampling method have affected the graduate study/employment rate
estimate from Rogers State University?
When writing about sampling bias…
1. Identify the population and the sample
2. Explain how the sampled individuals might __________________ the general population
3. Explain how this leads to an _________________________________.
Model Response:
“Graduates who didn’t find post-grad employment may be ashamed, making them ______________ to
respond to the survey. Therefore, this sampling method may include a lower proportion of unemployed
graduates than in the full population. This produces an _________________ of the true percentage of
_____ graduates who are actually starting full-time work.”
Important: These categories of bias can overlap. On an FRQ, if you’re unsure,
Types of selection bias
________ try to use one of these vocab terms. Instead, just describe the bias,
-Undercoverage bias
how it arises, and whether it leads to an under or overestimate.
-Nonresponse bias
-Voluntary response bias: Occurs when a sample is composed of _______________, who may differ
from individuals who don’t choose to volunteer.
Ex: You want to study heart rate during exercise. You recruit volunteers to run a mile and then
measure their pulse. The few insane people in our society who actually like to run are the ones
who volunteer, so they’re healthier on average than the population → bias
Types of survey bias
-Question wording bias: When survey questions are confusing or leading.
Ex: “Which show do you prefer: Diners, Drive-In, and Dives, hosted by the incredibly talented,
funny, and interminable mayor of Flavortown / chef Guy Fieri or Iron Chef hosted by the boring
Alton Brown?
-Self-reported response bias: When individuals inaccurately report their own traits.
Ex: I report being able to bench-press 350 lbs.
137
Simple Random Samples (SRS)
In order to avoid bias, you must _____________ sample.
Simple Random Sample (SRS): a sampling method in which every possible group of individuals in the
population has an __________________ of being selected.
Example: COVID-19 and Sampling
When COVID-19 spread to NYC, the city only provided tests to people who ____________________.
Some infected people don’t show symptoms. So, the sampling method led to an ________________ of
the number of people infected. It was __________.
Instead: We could have ______________________ the NYC population, tested those who were
sampled, and gotten an __________________ of the number of people infected.
Question: Describe how you would implement a simple random sample (SRS) of 1,000 NYC residents to
test for COVID.
When describing how to perform an SRS:
1. Assign each individual in the population a number 1 – N (population size).
2. Use a random number generator to obtain n (sample size) numbers, skipping repeats.
3. Sample the individuals whose numbers were generated
Model Response:
“Assign every individual in NYC an integer _______ (where N is the population size of NYC). Use a
random number generator to obtain 1,000 integers between 1 – N, __________________________.
Administer the COVID test to the 1,000 individuals whose numbers were selected.”
Lesson 4.1 Discussion
Discussion Question: During World War II, a statistician by the name of Abraham Wald was asked to
help the British air force decide where to put extra armor on their planes. They gave him charts of the
bullet holes in planes that were wounded in fighting but made it back safely to England. An example is
shown below, with each dot representing places hit by bullets.
Using the chart, on what part of a new plane would you
recommend they put extra armor? Choose from the
options below and give a statistical reason for your choice.
Options: A)Nose B)Wings C)Body D)Engine E)Tail
Image courtesy of Professor Joseph Blitzstein (i.e. the best stats prof in the country).
See his “Harvard Thinks Big” talk on this problem: https://youtu.be/dzFf3r1yph8
138
Does Beyoncé write her own lyrics?
1. Quickly circle a random sample of 5 words. Write them below. How many letters in
each word?
2. What is the average word length of your sample? ________.
3. Put your average on the dotplot on the white board at the front of the room. Copy the
class dotplot below.
4. Find a new sample of 5 words using a random number generator. Put your average on
the dotplot on the white board at the front of the room. Copy the class dotplot below.
5. How is the dotplot from #4 different than the dotplot for #3? Which do you think is a
better estimator of the true mean word length?
6. What do you think the true mean word length is for “Crazy in Love”?
7. It is known that Beyonce wrote the lyrics for all of the Destiny’s child songs. The
average word length for these songs is 3.64 letters. Based on your samples, do you
have good evidence that Beyonce did not write the lyrics for “Crazy in Love”. Explain.
139
Sampling Methods
Important ideas:
Check Your Understanding
1. A sign at a local business says, “Rate us with 5-stars on Google and get a $10 gift card!” The
store manager reads the ratings to learn how people in the community feel about the business.
a. What type of sample did the manager obtain?
b. Explain why this sampling method is biased.
c. The manager finds that 92% of customers rate the business with 5 stars. Is 92% likely to be
greater than or less than the percentage of all people in the community who truly believe the
business deserves 5-stars?
2. To help reduce bias, the manager decides to survey the next 10 customers that make a
purchase.
a. What type of sample did the manager obtain?
b. Explain why this sampling method is biased.
3. How could the manager avoid the bias described in Question 2?
140
Crazy in Love
I look and stare so deep in your eyes
I touch on you more and more every time
When you leave I'm begging you not to go
Call your name two or three times in a row
Such a funny thing for me to try to explain
How I'm feeling and my pride is the one to blame
'Cuz I know I don't understand
Just how your love can do what no one else can
Got me looking so crazy right now, your love's
Got me looking so crazy right now (in love)
Got me looking so crazy right now, your touch
Got me looking so crazy right now (your touch)
Got me hoping you'll page me right now, your kiss
Got me hoping you'll save me right now
Looking so crazy in love's
Got me looking, got me looking so crazy in love
When I talk to my friends so quietly
Who he think he is? Look at what you did to me
Tennis shoes, don't even need to buy a new dress
If you ain't there ain't nobody else to impress
The way that you know what I thought I knew
It's the beat my heart skips when I'm with you
But I still don't understand
Just how the love your doing no one else can
I'm Looking so crazy in love's
Got me looking, got me looking so crazy in love
Got me looking, so crazy, my baby
I'm not myself, lately I'm foolish, I don't do this
I've been playing myself, baby I don't care
'Cuz your love's got the best of me
And baby you're making a fool of me
You got me sprung and I don't care who sees
'Cuz baby you got me, you got me, so crazy baby
HEY!
141
1I
2 look
3 and
4 stare
5 so
6 deep
7 in
8 your
9 eyes
10 I
11 touch
12 on
13 you
14 more
15 and
16 more
17 every
18 time
19 When
20 you
21 leave
22 I'm
23 begging
24 you
25 not
26 to
27 go
28 Call
29 your
30 name
31 two
32 or
33 three
34 times
35 in
36 a
37 row
38 Such
39 a
40 funny
41 thing
42 for
43 me
44 to
45 try
46 to
47 explain
48 How
49 I'm
50 feeling
51 and
52 my
53 pride
54 is
55 the
56 one
57 to
58 blame
59 'Cuz
60 I
61 know
62 I
63 don't
64 understand
65 Just
66 how
67 your
68 love
69 can
70 do
71 what
72 no
73 one
74 else
75 can
76 Got
77 me
78 looking
79 so
80 crazy
81 right
82 now,
83 your
84 love's
85 Got
86 me
87 looking
88 so
89 crazy
90 right
91 now
92 (in
93 love)
94 Got
95 me
96 looking
97 so
98 crazy
99 right
100 now,
101 your
102 touch
103 Got
104 me
105 looking
106 so
107 crazy
108 right
109 now
110 (your
111 touch)
112 Got
113 me
114 hoping
115 you'll
116 page
117 me
118 right
119 now,
120 your
121 kiss
122 Got
123 me
124 hoping
125 you'll
126 save
127 me
128 right
129 now
130 Looking
131 so
132 crazy
133 in
134 love's
135 Got
136 me
137 looking,
138 got
139 me
140 looking
141 so
142 crazy
143 in
144 love
145 When
146 I
147 talk
148 to
149 my
150 friends
151 so
152 quietly
153 Who
154 he
155 think
156 he
157 is
158 Look
159 at
160 what
161 you
162 did
163 to
164 me
165 Tennis
166 shoes
167 don't
168 even
169 need
170 to
171 buy
172 a
173 new
174 dress
175 If
176 you
177 ain't
178 there
179 ain't
180 nobody
181 else
182 to
183 impress
184 The
185 way
186 that
187 you
188 know
189 what
190 I
191 thought
192 I
193 knew
194 It's
195 the
196 beat
197 my
198 heart
199 skips
200 when
142
201 I'm
202 with
203 you
204 But
205 I
206 still
207 don't
208 understand
209 Just
210 how
211 the
212 love
213 your
214 doing
215 no
216 one
217 else
218 can
219 I'm
220 Looking
221 so
222 crazy
223 in
224 love's
225 Got
226 me
227 looking,
228 got
229 me
230 looking
231 so
232 crazy
233 in
234 love
235 Got
236 me
237 looking,
238 so
239 crazy,
240 my
241 baby
242 I'm
243 not
244 myself,
245 lately
246 I'm
247 foolish,
248 I
249 don't
250 do
251 this
252 I've
253 been
254 playing
255 myself
256 baby
257 I
258 don't
259 care
260 'Cuz
261 your
262 love's
263 got
264 the
265 best
266 of
267 me
268 And
269 baby
270 you're
271 making
272 a
273 fool
274 of
275 me
276 You
277 got
278 me
279 sprung
280 and
281 I
282 don't
283 care
284 who
285 sees
286 'Cuz
287 baby
288 you
289 got
290 me,
291 you
292 got
293 me,
294 so
295 crazy
296 baby
297 HEY
Credit: Gabe
Yonker
4.1 PRACTICE!
1. In May 2015, the Los Angeles City Council voted to ban most travel and contracts with the state of Arizona to protest
Arizona’s new immigration enforcement law. The Los Angeles Times conducted an online poll that asked if the City
Council was right to pass a boycott of Arizona. The results showed that 96% of the 41,068 people in the sample said
“No.” Does this result represent the opinions of all Los Angeles residents? Explain.
2. The manager of a beach-front hotel wants to survey guests in the hotel to estimate overall customer satisfaction. The
hotel has two towers, an older one to the south and a newer one to the north. Each tower has 10 floors of standard
rooms (40 rooms per floor) and 2 floors of suites (20 suites per floor). Half of the rooms in each tower face the beach,
while the other half of the rooms face the street. This means there are (2 towers)(10 floors)(40 rooms) + (2 towers)(2
floors)(20 suites) = 880 total rooms.
A. Explain how to select a simple random sample of 88 rooms.
143
B. Explain how to select a stratified random sample of 88 rooms.
C. Explain why selecting 2 of the 24 different floors would not be a good way to obtain a cluster sample.
3. Which of the following are sources of sampling error and which are sources of nonsampling error? Explain your
answers.
(a) The subject lies about past drug use.
(b) A typing error is made in recording the data.
(c) Data are gathered by asking people to mail in a coupon printed in a newspaper.
144
4.2 Random Sampling Activity: Sampling Segregation
Most American cities have severe economic segregation. This means that households in the same
neighborhood have similar incomes, but incomes across neighborhoods tend to be quite different.
Multiple reports have found that San Antonio is the most economically segregated city in the United
States. The following activity investigates the challenges of sampling incomes in San Antonio.
Judgement Sample: Select 5 homes that you think are representative of the whole population
of San Antonio. Label the home numbers on the bottom lines and their incomes on the top
lines. Calculate the mean of these incomes and record your answer in the space provided.
Incomes:
________
___
Home Number:
________
___
_________
___
________
___
________
___
Mean: _______
Simple Random Sample (SRS): Use a random number generator to select 5 homes numbered
between 1 and 100. Label the home numbers on the bottom lines and their incomes on the top
lines. Calculate the mean of these incomes and record your answer in the space provided.
Incomes:
________
___
Home Number:
________
___
_________
___
________
___
________
___
Mean: _______
Cluster Random Sample: Use a random number generator to select 1 home numbered
between 1 and 100. Find the home on the map as well as the four homes located closest to it.
Label the home numbers on the bottom lines and their incomes on the top lines. Calculate the
mean of these incomes and record your answer in the space provided.
Incomes:
________
___
Home Number:
________
___
_________
___
________
___
________
___
Mean: _______
Stratified Random Sample: Find the page with the title “Strata by Race-Ethnicity”, in which the
homes are divided by predominate race. Use a random number generator to select 1 home
within each of the 5 strata. For example, I would use RandInt(67,71,1) to select a Native
American home, as the range for that strata is 67-71. Label the home numbers on the bottom
lines and their incomes on the top lines. Calculate the mean of these incomes.
Incomes:
________
___
Home Number:
________
___
_________
___
________
___
________
___
Mean: _______
Systematic Random Sample: Use a random number generator to select a number between 1
and 20. This is the first home. Then add 20 to each previous number to find the other four
homes. Label the home numbers on the bottom lines and their incomes on the top lines.
Calculate the mean of these incomes.
Incomes:
________
___
Home Number:
________
___
_________
___
145
________
___
________
___
Mean: _______
94
93
42
49
91
41
87
50
88
90
89
92
86
40
95
57
82
43
81
45
48
98
56
46
83
60
79
58
55
75
78
76
33
18
74
35
44
59
84
80
77
96
85
54
47
34
11
17
99
8
71
32
97
63
9
13
5
72
6
61
1
7
2
19
20
53
52
25
70
26
23
21
68
27
22
29
51
15
64
66
4
67
100
24
3
65
14
62
10
16
12
73
28
69
31
30
Key (Predominate Race in Home):
37
38
Asian
Black
Hispanic
Native American
White
39
36
Home Income
Home Income
Home Income
Home Income
Home Income
$25,000
Home Income
Home Income
1
16
$23,500
31
$64,000
46
$65,000
61
$66,500
76
$56,000
91
$108,000
2
$26,500
17
$23,500
32
$35,000
47
$62,500
62
$26,000
77
$76,000
92
$157,000
3
$42,000
18
$53,000
33
$55,000
48
$78,000
63
$24,500
78
$44,500
93
$186,000
4
$56,000
19
$36,000
34
$56,500
49
$85,500
64
$22,000
79
$212,500
94
$166,000
5
$23,000
20
$28,500
35
$57,000
50
$61,500
65
$20,000
80
$119,000
95
$174,000
6
$24,500
21
$27,500
36
$42,500
51
$48,000
66
$28,500
81
$67,000
96
$80,000
7
$44,500
22
$28,000
37
$44,500
$23,000
82
$124,000
97
$42,000
8
$25,500
23
$30,500
38
$41,000
$20,000
83
$109,000
98
$48,500
9
$25,500
24
$27,000
39
$43,000
$27,500
84
$111,500
99
$28,000
10
$54,000
25
$31,500
40
$109,000
55
$44,500
70
$36,000
85
$48,000
100
$27,000
11
$26,000
26
$29,500
41
$214,500
56
$102,000
71
$25,000
86
$43,500
12
$25,000
27
$45,500
42
$92,500
57
$139,000
72
$57,000
87
$47,000
13
$35,500
28
$61,000
43
$95,000
58
$68,000
73
$110,000
88
$39,000
14
$25,000
29
$61,000
44
$68,500
59
$107,500
74
$28,000
89
$58,500
15
$27,500
30
$58,500
45
$76,000
60
$85,000
75
$58,000
90
$62,000
52
53
54
$44,500
$47,000
$108,500
146
67
68
69
Strata: By Race-Ethnicity
Asian
57
56
58
Black
61
59
65
62
66
63
60
64
Hispanic
1
6
11
2
7
12
3
16
21
17
22
18
23
13
8
48
28
14
9
19
15
33
54
45
24
20
49
41
37
38
50
34
30
25
10
5
53
44
36
32
29
4
52
35
31
27
47
43
40
42
39
55
46
51
26
Native American
67
68
71
69
70
White
72
82
79
76
90
85
96
93
86
91
97
83
73
77
99
80
87
98
92
74
75
78
84
81
100
94
88
95
89
147
Sampling Distributions for Each Sample Type
Judgement Sample
---------|------------|------------|------------|------------|------------|------------|------------|------------|-------$20k
$30k
$40k
$50k
$60k
$70k
$80k
$90k
$100k
Simple Random Sample (SRS)
---------|------------|------------|------------|------------|------------|------------|------------|------------|-------$20k
$30k
$40k
$50k
$60k
$70k
$80k
$90k
$100k
Cluster Random Sample
---------|------------|------------|------------|------------|------------|------------|------------|------------|-------$20k
$30k
$40k
$50k
$60k
$70k
$80k
$90k
$100k
Stratified Random Sample
---------|------------|------------|------------|------------|------------|------------|------------|------------|-------$20k
$30k
$40k
$50k
$60k
$70k
$80k
$90k
$100k
Systematic Random Sample
---------|------------|------------|------------|------------|------------|------------|------------|------------|------$20k
$30k
$40k
$50k
$60k
148
$70k
$80k
$90k
$100k
AP Statistics: Lesson 4.2 Post-Activity Guided Notes & Discussion
Topics: random sampling methods
High accuracy → unbiased
Low accuracy → biased
High precision → low variability
Low precision → high variability
Judgement Sample
Biased (not accurate) / Unbiased (accurate)
Precise (low variability) / Imprecise (high variability)
Pros:
Cons:
Question: Why was the judgement sample biased?
Simple Random Sample (SRS)
Biased (not accurate) / Unbiased (accurate)
Precise (low variability) / Imprecise (high variability)
Pros:
Cons:
Cluster Random Sample
Biased (not accurate) / Unbiased (accurate)
Precise (low variability) / Imprecise (high variability)
Pros:
Cons:
Question: Why did the cluster random sample have high variability?
Stratified Random Sample
Biased (not accurate) / Unbiased (accurate)
Precise (low variability) / Imprecise (high variability)
Pros:
Cons:
Question: Why did the stratified random sample have low variability?
Systematic Random Sample
Biased (not accurate) / Unbiased (accurate)
Precise (low variability) / Imprecise (high variability)
Pros:
Cons:
149
Lesson 4.2 Discussion – Property Taxes and Financing Schools
Alamo Heights High School (Alamo Heights ISD)
•
•
•
Property tax rate: $1.06 per $100 in home value
Average home value: $560,500 (source: Zillow)
Calculate tax revenue from average home:
Burbank High School (San Antonio ISD)
•
•
•
Property tax rate: $1.17 per $100 in home value
Average home value: $125,000 (source: Zillow)
Calculate tax revenue from average home:
Discussion Question: Funding for schools varies by state, county, and township. If you were mayor of
San Antonio, based on the activity and your analysis above, which of the following school funding
schemes would you choose? Why?
Option A: Fund schools based only on property taxes collected from the homes around the school.
Option B: Fund all schools equally (the same amount of money per student) by taking tax money from
property-rich areas and giving it to schools in property-poor areas.
Option C: Give more funds per student to schools that serve lower-income students. You do this by
taking more tax money from property rich areas and giving it to schools in property-poor areas.
150
4.2 PRACTICE!
1. A study published in the New England Journal of Medicine compared two medicines to treat head lice: an oral
medication called ivermectin and a topical lotion containing malathion. Researchers studied 812 people in 376
households in seven areas around the world. Of the 185 randomly assigned to ivermectin, 171 were free from head lice
after two weeks compared to only 151 of the 191 households randomly assigned to malathion.
Identify the experimental units, explanatory and response variables, and the treatments in this experiment.
2. Does adding fertilizer affect the productivity of tomato plants? How about the amount of water given to the plants?
To answer these questions, a gardener plants 24 similar tomato plants in identical pots in his greenhouse. He will add
fertilizer to the soil in half of the pots. Also, he will water 8 of the plants with 0.5 gallons of water per day, 8 of the
plants with 1 gallon of water per day and the remaining 8 plants with 1.5 gallons of water per day. At the end of three
months he will record the total weight of tomatoes produced on each plant.
Identify the explanatory and response variables, experimental units, and list all the treatments.
151
3. Suppose you have a class of 30 students who volunteer to be subjects in an experiment involving caffeine. Explain
how you would randomly assign 15 students to each of the two treatments.
4. A cell phone company is considering two different keyboard designs (A and B) for its new line of cell phones.
Researchers would like to conduct an experiment using subjects who are frequent texters and subjects who are not
frequent texters. The subjects will be asked to text several different messages in 5 minutes. The response variable will
be the number of correctly typed words.
A. Explain why a randomized block design might be preferable to a completely randomized design for this experiment.
B. Outline a randomized block experiment using 100 frequent texters and 200 novice texters.
152
How Much Do Fans Love Justin Timberlake? Day 1
Justin Timberlake’s concert promoter wants to find out how much fans enjoy the concerts. He
will ask fans, “From 1 to 100, where 100 is the most, how much did you enjoy the concert?” The
section he wants to survey has 50 seats (5 rows x 10 columns). The stage runs along the
northern edge of the venue (where Justin is pictured). He wants to take a sample of 10 seats.
1. Method #1:
Take a simple random sample (SRS) of 10
fans. Explain below the steps you used to
obtain an SRS.
2. Method #2:
Randomly choose 2 fans from each
horizontal row.
3. Method #3:
Randomly choose 1 fan from each
vertical column.
4. Which method do you think is best? Why?
153
5. Now, it’s time for the actual data. For each of your samples on the previous page,
calculate the average enjoyment. Add your average to the dotplots on the board.
Sample #1:
Sample #2:
Sample #3:
Method #1: SRS
average enjoyment
Method #2: Stratify by Row
average enjoyment
Method #3: Stratify by Column
average enjoyment
154
Other Random Sampling Methods Day 1
Important Ideas:
Check Your Understanding:
To score the AP Statistics Exams ETS hires Exam Readers, Table Leaders, and Other
Leadership. Each reading room consists of 16 Exam Readers and 2 Table Leaders. There are
100 reading rooms. The 18 members of Other Leadership work together in a room.
a. Describe how to select a stratified random sample of 36 people hired by ETS to score
the AP Statistics Exams. Explain your choice of strata.
b. Describe how to select a cluster sample of 36 people hired by ETS to score the AP
Statistics Exams. Explain your choice of clusters.
c. Explain a benefit of using a stratified random sample and a benefit of using a cluster
random sample in this context.
155
AP Statistics Handout: Lesson 4.3
Topics: observation vs. experiment, components and principles of experiments, completely
randomized design
Lesson 4.3 Guided Notes
Observational Studies vs. Experiments
Sources:
-Huang, J. et al. “Historical comparison
of gender inequality in scientific
careers across countries and
disciplines.” Proceedings of the
National Academy of Sciences, Mar
2020, 117 (9) 4609-4616; DOI:
10.1073/pnas.1914221117
-Boston Consulting Group, “What’s
Keeping Women out of Data Science?”
bcg.com/publications/2020/whatkeeps-women-out-data-science.aspx
1. One possible cause of gender gaps is hiring bias. Provide an explanation for how hiring biases might
cause these trends:
From this ____________ data alone, we cannot prove that hiring discrimination is the cause. Why?
Confounding variables: Provide ___________________ explanations for trends between explanatory
(gender) and response (hiring rates) variables.
2. Name and discuss one confounding variable that could also explain the gender gap in STEM:
Observational study: a study in which data is collected ___________________________ any treatments.
• Retrospective study: examines ________________________ on individuals
• Prospective study: follows individuals to gather ____________________
• Both ______________ show cause and effect because they do not control for confounding!
156
Experiments: a study in which treatment is _______________ on subjects.
• If well designed, experiments _____________ cause-effect relationships by _________________
for confounding variables.
Components of Experiments
The Jenn/John Study - Example inspired by Advanced High School Statistics
In this study*, experimenters printed up copies of fake application materials for a science lab manager
job. All copies of the application materials were completely identical, except for one thing: the name. On
about half of the copies, the application listed the name “John.” The other half had the name “Jennifer.”
Experimenters found 127 science lab faculty members and randomly sent them either a “John” (n = 63)
or “Jennifer” (n = 64) application. Each faculty member was told that the application was for a real
position at their University. The faculty members independently rated the applicant’s “hireability” (1-7
scale) and estimated their starting salary. Experimenters compared these results across the two groups.
*Moss-Racusin, C., Dovidio, J., et al. “Science faculty’s subtle gender biases favor male students.”
PNAS October 9, 2012 109 (41) 16474-16479; https://doi.org/10.1073/pnas.1211286109
Experimental units: the ________________________ (person, animal, plant, virus, particle, etc.) that are
assigned to different treatments.
In hiring study: _______________________
Explanatory variable: the variable that is purposefully __________________. This is also known as the
factor.
In hiring study: ____________________________
Treatments: the different _____________ of the explanatory variable in the experiment.
In hiring study: __________________
Response variable: the measured experiment ___________________ that is compared between
treatment groups.
In hiring study: ___________________
157
Principles of Experimental Design
•
•
•
•
________________ of at least two treatment groups
_____________________________ of experimental units to treatment
_____________________– many experimental units in each treatment group
________________ of confounding variables
1. Comparison
a) Explain how comparison is implemented in the Jenn/John study:
2. Random Assignment
a) Why would it be a problem if I assigned Tier 1 University labs to get “John” and Tier 2
labs to get “Jennifer” application materials?
b) Random assignment tends to ________________ confounding factors, so inferences can be made
about the explanatory variable.
c)
Sampling
Random ___________
→ Reduces ________
Experiments
Random ____________
→ Reduces _______________
3. Replication
a) Why would it be a problem if I only assigned one faculty member to the “John” group and one
faculty member to the “Jennifer” group?
b) Larger treatment group size reduces the likelihood of differences arising due to chance alone. In
other words, it makes our estimates of treatment effect _______________________.
4. Control
a) One way they control for confounding factors is by making the application materials ____________
_______________, except for the explanatory variable (gender).
158
Describing a Completely Randomized Design
Completely randomized design: An experimental design in which experimental units are assigned to
treatments _______________________________________.
• This is the “SRS” of experiments – the simplest (but still effective) randomized experiment.
Question: Describe how you would implement a completely randomized design of the Jennifer/John
experiment, with 127 science faculty members.
When describing how to perform a completely randomized design experiment:
1. Assign each experimental unit a number 1 – n (sample size).
2. Write all the numbers on identical slips of paper, put into a hat, and mix well.
3. Draw out nt (treatment group size) slips of paper, without replacement. The corresponding units are
assigned treatment 1. Draw out another nt slips of paper, assign to treatment 2, etc.
4. Compare response among treatment groups
Model Response:
“Assign each faculty member an integer, __________. Write integers 1-127 on identical slips of paper,
put them into a hat, and _______________. Draw out 63 slips (without replacement). The
corresponding faculty members will receive ‘John’ application materials. The __________________
faculty members will receive ‘Jennifer’ application materials. At the end of the experiment, record
faculty members’ rating of applicants’ ‘hireability’ and starting salary estimates. Finally,
_______________ these results across the two groups.”
Lesson 4.3 Discussion
Here’s an estimated summary of the Jenn/John study results:
Estimates based on
publicly available data
from the study.
The differences were found to be statistically significant:
so extreme that they were unlikely to happen by
__________________.
Hireability
Salary
John
3.78
$30,238
Jennifer
2.93
$26,508
Since this was a well-designed experiment, we can infer that these average differences are
_________________ by gender bias.
159
Difference
-0.85
-$3,730
How Much Do Fans Love Justin Timberlake? Day 2
In the next city, Justin Timberlake’s concert promoter again wants to find out how much fans
enjoy his concerts. He will ask fans, “From 1 to 100, where 100 is the most, how much did you
enjoy the concert?” Again, he wants to take a sample of 10 fans. He also would like to try out a
couple of new methods for sampling.
1. Method #1:
Take a simple random sample (SRS) of 10
fans.
2. Method #2:
To make it easier to distribute the surveys, the
promoter decides to pick one row and just
sample every fan in that row.
a.
Use this method to select a sample of 10
fans.
b.
Do you think this method will produce
good estimates? Why or why not?
3. Method #3:
Justin’s manager thinks it is important to
sample fans that have different views of the
stage. He wants to sample every 8th fan.
a.
First, we need to figure out the starting
fan. Randomly select a fan and mark with
an X.
b.
Begin marking every 8th seat until you get a
sample of 10 seats (start back at the
beginning if you need to).
4. Which method do you think is best? Why?
160
5. Now, it’s time for the actual data. For each of your samples on the previous page,
calculate the average enjoyment. Add your average to the dotplots on the board.
Sample #1:
Sample #2:
Sample #3:
Method #1: SRS
average enjoyment
Method #2: Cluster Sample
average enjoyment
Method #3: Systematic Random Sample
average enjoyment
161
More Sampling Methods Day 2
Important Ideas:
Check Your Understanding:
The manager of a large ocean front hotel would like to survey their guests to determine their
satisfaction with the view from their room. The hotel has 10 floors. Half of the rooms overlook the
ocean and the other half overlook the street. There are 50 rooms on each floor, for a total of 500
rooms. The hotel manager would like to select a sample of 50 rooms.
a. Describe how to select a stratified random sample of 50 rooms.
b. Describe how to select a cluster sample of 50 rooms.
c. Describe how to select a systematic random sample of 50 rooms.
d. Explain a benefit of using each of the three types of sampling methods in this context.
162
AP Statistics Handout: Lesson 4.4
Topics: blocking, matched pairs, placebo, blinding, generalizing study results
Lesson 4.4 Guided Notes
Blocking
The Jenn/John Study - Example inspired by Advanced High School Statistics
In this study*, experimenters printed up copies of fake application materials for a science lab manager
job. All copies of the application materials were completely identical, except for one thing: the name. On
about half of the copies, the application listed the name “John.” The other half had the name “Jennifer.”
Experimenters found 127 science lab faculty members and randomly sent them either a “John” (n = 63)
or “Jennifer” (n = 64) application. Each faculty member was told that the application was for a real
position at their University. The faculty members independently rated the applicant’s “hireability” (1-7
scale) and estimated their starting salary. Experimenters compared these results across the two groups.
*Moss-Racusin, C., Dovidio, J., et al. “Science faculty’s subtle gender biases favor male students.”
PNAS October 9, 2012 109 (41) 16474-16479; https://doi.org/10.1073/pnas.1211286109
The experiment described above is a…
Completely randomized design: An experimental design in which experimental units are assigned to
treatments completely at random.
• This is the ________ of experiments – the simplest (but still effective) randomized
experiment.
What if the faculty members were from both Tier 1 and Tier 2 Universities?
There is some difference but also
some overlap. The differences may
be due to _____________________.
163
When broken down by Tier, there is
little overlap. Schools ____________
each tier rated John more ‘hireable.’
How to reduce the variability due to school tier –
Randomized Complete Block Design: experimental units are first blocked (_________________) by a
similar trait that may affect response. Then, units from each block are randomly assigned to treatment.
• This is the _______________________________ of experiments.
• It reduces variation between treatment groups at the start of the experiment. This makes it
easier to show that differences in response are really _______________________________,
rather than chance variation in the random assignment.
Question: Describe how you would implement a randomized complete block design for the Jenn/John
study, which takes into account University tier.
Model Response:
“Block the science lab faculty members by University tier. In each block, assign each faculty member
____________________________. For the Tier 1 block, write all the numbers on identical slips of paper,
put into a hat, and mix well. Draw out 3 slips of paper, _____________________________________.
The corresponding faculty are assigned ‘Jennifer’ application materials. The remaining are assigned
‘John’ materials. _______________ the same random assignment process for the Tier 2 block. Compare
‘hireability’ ratings between the Jenn/John groups ____________ each block. ____________________
the results, after accounting for the average difference in each block.”
164
Matched Pairs Design
Matched Pairs Design: a type of randomized blocked experiment in which each block is composed of
_______ similar experimental units (a “matched pair”).
• Often, the “matched pair” is simply the same experimental unit receiving both treatments. The
____________ of the treatments is randomized.
The 2015 Journal of the American Medical Association Depression Study
In this study*, researchers wanted to test if taking a “fake pill” would actually alleviate depression
symptoms, even though the pill had no active ingredients. 35 people enrolled in the study. All had major
depression and none were taking any medications. Because depression varies greatly between
individuals, the researchers implemented a matched pairs design.
*Simplified version of: Association Between Placebo-Activated Neural Systems and Antidepressant Responses: Neurochemistry
of Placebo Effects in Major Depression. Peciña M, Bohnert AS, Sikora M, Avery ET, Langenecker SA, Mickey BJ, Zubieta JK. JAMA
Psychiatry. 2015 Sep 30:1-8. doi: 10.1001/jamapsychiatry.2015.1335. [Epub ahead of print]. PMID: 26421634.
Question: Describe how you would implement a matched pairs design for the depression study.
“For each subject, flip a coin. Heads indicates they get the
fake pill for the first week, no pill for the second week. Tails
indicates the _____________________________________.
Have each subject fill out a depression questionnaire and
undergo a PET brain scan at the end of each week. Compare
measurements _________________________ and compile
the results.”
Depression study results
1. On average, participants reported less severe depressive symptoms after their week with
the fake pill.
2. On average, participants showed “increased µ-opioid receptor brain activity in regions of the
brain associated with emotion and stress regulation.”
o Belief in the pill caused a ____________________________
“Fake pills” have also have also shown significant beneficial effects for….
• Migraines
• Blood Pressure
Called the:
• Asthma
_______________________
• Arthritis
• Many other ‘physical’ illnesses
Source: https://www.ncbi.nlm.nih.gov/books/NBK513296/
165
Blinding and Placebo
Placebo: An ________________ treatment (e.g. sugar pill or salt water
IV drip)
Placebo effect: when subjects’ belief of receiving an active treatment
leads to a measured response, even though the treatment is
___________________________.
Single-blind study: _______________ the subjects or the researchers are unaware of who receives
active treatment or placebo.
Double-blind study: ________ the subjects and the researchers are ___________________ of who
receives active treatment or placebo.
Why blind subjects? _____________________________
Why blind researchers? ___________________________
• Researchers trying to prove that a treatment works may favor treatment group in their
measurements
Lesson 4.4 Discussion
Here’s an estimated summary of the Jenn/John study results:
Hireability
Salary
John Jennifer Difference
3.78
2.93
-0.85
$30,238 $26,508 -$3,730
Estimates based on publicly
available data from the study.
Generalization: using study results to make inferences about a _________________________.
Discussion Question: In the Jenn/John study, all faculty participants were from one of three science
departments: biology, chemistry, or physics. Can we generalize the result of gender hiring discrimination
to all scientific subjects? Explain your reasoning.
166
Were subjects randomly sampled?
Generalization and Scope of Inferences
Were subjects randomly assigned to treatment?
Subjects were
randomly
sampled from
the population
Subjects were
not randomly
sampled from
the population
Subjects were randomly assigned to
groups (experiment)
Subjects were not randomly assigned
to groups (observational study)
Generalize to the population: ___
Determine cause and effect: ___
Generalize to the population: ___
Determine cause and effect: ___
Generalize to the population: ___
Determine cause and effect: ___
Generalize to the population: ___
Determine cause and effect: ___
167
What is wrong with these surveys?
Identify what is wrong in each of these surveys. Be sure to explain.
1. The mayor of Springfield is interested in finding out the average age of people in the city.
He obtains a list of all of the landline telephones in the city, and then contacts a simple
random sample of 300 people. He uses the data from the sample to estimate the average
age of all the people in the city.
a. What is wrong with this survey?
b. Do you think the Mayor will over or underestimate the true mean age of people in
Springfield? Why?
2. The administration at a school wants to know the proportion of students that did all of
their homework last night. They select a simple random sample of 100 students and send
an email to each of them asking if they did all of their homework last night. Of the 40
responses, 36 of the students said that they did all of their homework last night (90%).
a. What is wrong with this survey?
b. Do you think the administration will over or underestimate the true proportion of
students who did all of their homework last night? Why?
3. Boy Scout Peter M. wants to know the proportion of people in his neighborhood who
support the Boy Scouts. He takes a random sample of 30 homes and visits them dressed
in his uniform.
a. What is wrong with this survey?
b. Do you think Peter will over or underestimate the true proportion of his neighbors
who support the Boy Scouts? Why?
168
Sample Surveys: What else can go wrong?
Important Ideas:
Check Your Understanding:
1. The principal of a large high school wants to learn about student opinion regarding remote
learning versus in person learning. Each of the following methods introduces a possible
source of bias. Name the type of bias.
a. The principal selects a random sample of students from those who signed up for remote
learning.
b. The principal emails a random sample of 100 students, but only 28 students respond.
c. The principal surveys the first 50 students that submit their learning preference for the
upcoming school year.
2. The following question was on the Pennsylvania General Election Ballot in the 2016 election.
“Shall the Pennsylvania Constitution be amended to require that justices of the
Supreme Court, judges, and magisterial district judges be retired on the last day of
the calendar year in which they attain the age of 75 years?”
The wording seems to ask if there should be mandatory retirement. However, what was not
revealed is that mandatory retirement already existed at age 70. Due to the wording of the
question is the percentage of people who voted “Yes” likely to be less than, greater than, or
about equal to the percent of all voters who would vote “Yes” if the question were not
misleading? Explain your reasoning.
169
Does SAT prep improve scores?
Last year EKHS offered an after school SAT prep class that students could volunteer to take.
44 students took the course and then took the SAT. The average SAT score for this group was
1220. The average SAT score for all students who did not take the prep class was 1050.
1. Is the situation described an observational study or an experiment?
2. Identify the explanatory variable and the response variable.
3. Can you conclude that taking the prep course will cause a student’s SAT score to
increase? Why or why not?
4. Identify as many other possible variables that you can that may explain why the
SAT scores are higher for those who took the prep course than for those who did
not.
5. Design an experiment that would allow us to determine if the SAT prep causes
an increase in SAT scores. Be as thorough as possible.
170
Observational Studies and Experiments
Important Ideas:
Check Your Understanding:
1. Do homes with metal roofs get worse cell service than homes with shingled roofs? Ben and
Jerry are neighbors and own an ice cream business together. Ben has a metal roof on his
home and Jerry has a shingled roof. Ben and Jerry select a random sample of 50 employees.
They randomly assign them to come to a party at either Ben’s house or Jerry’s house. When
they arrive Ben and Jerry ask them about their signal strength. Was this an observational study
or an experiment? Justify your answer.
2. A study showed that one factor that strongly correlates with student academic success is the
number of books that the family has in their home. Researchers randomly selected 100
students and measured their academic success as well as counted the number of books their
family has in their home.
a. Is this an observational study or an experiment? Justify your answer.
b. What are the explanatory and response variables?
c. Explain clearly why such a study cannot establish a cause-and-effect relationship. Suggest
a variable that may be confounded with number of books that the family have in their home.
171
Would you fall for that?
Would you fall for the placebo effect? Watch this video, then complete the rest of the questions.
1. Why do you think the people in the video got stronger?
Similar to the video, Mrs. Gallas wants to use a beverage to test the affect that caffeine can have on
heart rate. Here is an initial plan:
• measure initial pulse rate
• give each student some caffeine (Coca-Cola)
• wait for a specified time
• measure final pulse rate
• compare final and initial rates
2. What are some problems with this plan? What other variables will be sources of variability in
pulse rates?
3. Go back up to your list in #2 and propose a solution to each problem.
4. Design an experiment to test the effect that caffeine has on heart rate.
172
Designing Experiments
Important Ideas:
Check Your Understanding:
A group of researchers in Africa find a creative way to protect cattle from lion attacks – they paint eyes on
the cows’ rears. To determine if this treatment is effective, they randomly assign the cattle to one of three
treatments: Eyes on their rear, cross-marks on their rear, or nothing on their rear. After 4 years of roaming
the plains, the cows with eyes saw no deaths, the cows with a cross suffered 4 deaths, and the cows with
no marks suffered 15 deaths.
a. Explain why it was important to have a control group that didn’t get a mark on their rear.
b. Suppose the researchers had 1200 cows. Describe how to randomly assign the cows to the treatments.
c. What is the purpose of randomly assigning treatments in this context?
d. Create an outline showing a completely randomized design for the experiment.
173
4.2 PRACTICE!
1. A study published in the New England Journal of Medicine compared two medicines to treat head lice: an oral
medication called ivermectin and a topical lotion containing malathion. Researchers studied 812 people in 376
households in seven areas around the world. Of the 185 randomly assigned to ivermectin, 171 were free from head lice
after two weeks compared to only 151 of the 191 households randomly assigned to malathion.
Identify the experimental units, explanatory and response variables, and the treatments in this experiment.
2. Does adding fertilizer affect the productivity of tomato plants? How about the amount of water given to the plants?
To answer these questions, a gardener plants 24 similar tomato plants in identical pots in his greenhouse. He will add
fertilizer to the soil in half of the pots. Also, he will water 8 of the plants with 0.5 gallons of water per day, 8 of the
plants with 1 gallon of water per day and the remaining 8 plants with 1.5 gallons of water per day. At the end of three
months he will record the total weight of tomatoes produced on each plant.
Identify the explanatory and response variables, experimental units, and list all the treatments.
174
3. Suppose you have a class of 30 students who volunteer to be subjects in an experiment involving caffeine. Explain
how you would randomly assign 15 students to each of the two treatments.
4. A cell phone company is considering two different keyboard designs (A and B) for its new line of cell phones.
Researchers would like to conduct an experiment using subjects who are frequent texters and subjects who are not
frequent texters. The subjects will be asked to text several different messages in 5 minutes. The response variable will
be the number of correctly typed words.
A. Explain why a randomized block design might be preferable to a completely randomized design for this experiment.
B. Outline a randomized block experiment using 100 frequent texters and 200 novice texters.
175
Does type of SAT prep matter?
EKHS has decided to offer an SAT prep class again this year. It will be offered in two different
formats: online or classroom teacher. The counselors want to know which teaching method will
yield higher SAT scores so they have allowed us to set up an experiment. 50 students have
signed up to take some form of the SAT prep class. (20 seniors and 30 juniors)
1. Outline a completely randomized design to compare the two treatments.
2. The counselors at EKHS hypothesize that the online vs. classroom results could
be greatly affected by the grade level of students that were put into each
treatment group. They know that seniors generally score better on the SAT than
juniors. How could we adjust our experiment to ensure that there is even split of
seniors and juniors in each class? Draw an outline of the experiment with your
modifications.
3. The counselors are now worried that a student’s GPA is certainly going to affect
their SAT score. Let’s look only at the Juniors. We want to be sure that the
different GPAs are being evenly distributed into the two treatment groups.
How could we be sure the GPAs are evenly distributed?
176
Randomized Block Designs
Important Ideas:
Check Your Understanding:
A political strategist would like to design an experiment to compare the effectiveness of
three different YouTube advertisements promoting a specific presidential candidate.
She will use 300 randomly selected YouTube users for the experiment.
1. Describe a completely randomized design to compare the effectiveness of the
three advertisements.
2. Describe a randomized block design for this experiment. Justify your choice of
blocks.
3. Why might a randomized block design be preferable in this context?
177
Lesson 4.3: Day 1: Does caffeine increase pulse rate?
Mrs. Gallas and her students decided to perform the caffeine experiment. In their experiment, 10
student volunteers were randomly assigned to drink cola with caffeine and the remaining 10
students were assigned to drink caffeine-free cola. Were their findings statistically significant?
The table shows the change in pulse rate for each student (Final pulse rate – Initial pulse rate),
along with the mean change for each group.
1. Find the difference in mean pulse rate for the groups. Does your initial reaction lead you to
believe that they found evidence that caffeine does or does not increase heart rate? Explain.
2. What are two possible explanations for the difference in mean pulse rate?
To try to decide if the difference in pulse rate is big enough to be convincing we will do a simulation
with the data.
Simulation:
Step 1: Gather 20 index cards to represent the 20 students in this experiment. On each card, write
one of the 20 outcomes listed in the table. For example, write “8” on the first card, “3” on the second
card, and so on.
Step 2: Shuffle the cards and deal two piles of 10 cards each. This represents randomly assigning
the 20 students to the two treatments, assuming that the treatment received doesn’t affect the
change in pulse rate. The first pile of 10 cards represents the caffeine group, and the second pile of
10 cards represents the no-caffeine group.
Step 3: Fill in the table below with your simulated data.
Caffeine
No
Caffeine
178
3. Find the mean change for each group in your simulation and subtract the means (Caffeine –
No caffeine).
4. Add your difference in means to the dotplot on the board. Sketch the dotplot below.
What does each dot represent?
5. What percentage of the dots are greater than or equal to the difference in means of 1.2 found
in Mrs. Gallas’ experiment?
Interpret this percentage:
6. Do you think the difference in means we found from our experiment is due to the caffeine or
has it occurred purely by chance? Explain.
179
Lesson 4.3: Day 1: Inference for Experiments
Big Ideas:
Check Your Understanding:
1. How much do National Football League (NFL) players weigh, on average? In a random
sample of 50 NFL players, the average weight is 244.4 pounds.
(a) Do you think that 244.4 pounds is the true average weight of all NFL players? Explain your
answer.
(b) If another random sample of 50 NFL players was taken, would you expect to an average
weight of exactly 244.4 pounds?
(c) Estimates are usually given with a margin of error. The margin of error for the estimate of
244.4 pounds is 14.2 pounds. Based on this, would you be surprised if the true average
weight of NFL players was 260 pounds? Explain.
(d) Which would be more likely to give an estimate close to the true average weight of all NFL
players: a random sample of 50 players or a random sample of 100 players? Explain your
answer.
180
What’s in a name?
Your class just performed a modified version of the famous “John/Jenn” study* conducted by
researchers at Yale University. You will now perform calculations with the resulting data and
investigate if your findings are statistically significant.
*Moss-Racusin, C., Dovidio, J., et al. “Science faculty’s subtle gender biases favor male students.” PNAS October 9,
2012 109 (41) 16474-16479; https://doi.org/10.1073/pnas.1211286109
Surprise! The resumé you evaluated was fake. You just took part in an experiment. Each resumé the
class evaluated was completely identical except for one component: the first name of the applicant.
Half of the class received a resumé with the name “John Miller,” and the other half received one for
“Jennifer Miller.” Let’s see if the name, alone, was enough to influence your class’s ‘hireability’
estimates.
Ratings for John
Ratings for Jennifer
John mean rating:
Jennifer mean rating:
Difference (John – Jenn):
Is the difference between these mean ratings statistically significant? To decide this, we will do a
simulation with the data.
Simulation:
Step 1: We will use index cards to represent the different ratings in this experiment. On each card,
write one of the ratings listed in the above tables (for both John and Jennifer). For example, if there
were eight ratings of ‘3’ above, then eight index cards should have a ‘3’ on them.
Step 2: Shuffle the cards and deal two even piles. This represents randomly assigning the ratings to
the two name groups (assuming gender doesn’t affect the rating). The first pile of cards represents
the “John” group, and the second pile cards represents the “Jennifer” group.
Step 3: Fill in the table below with your simulated data.
John
Jennifer
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
181
1. Find the mean for each group in your simulation and subtract the means (John - Jenn).
2. Add your difference between means to the dotplot on the board. Sketch the dotplot below.
What does each dot represent?
3. What percentage of the dots are greater than or equal to the difference in means we found in
our experiment?
Interpret this percentage:
4. Do you think the difference between mean ‘hireability’ ratings we found from our experiment
is due to the name or has it occurred purely by chance? Explain.
5. In the original Yale study, applications were sent to STEM faculty at various Universities for a
lab manager position. Among the 127 faculty that were randomly assigned either Jenn/John
materials, the mean ‘hireability’ rating for John was 3.78 and for Jenn was 2.93. Is this
evidence more convincing or less convincing of gender bias than the data from our class
experiment? Explain.
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
182
Inference for Experiments
Important Ideas:
Check Your Understanding:
The following is an example of A/B testing, which is a common data science practice among tech and
marketing companies. The marketing team at Yelp came up with two new formats for displaying ads
on its site - format A and format B. Before deciding on which format to use across its entire site, Yelp
wants to test which format would get a higher rate of ad clicks from users. To test this, the company
randomly assigns 50 user accounts to see format A and 50 user accounts to see format B. After an
hour, format A had 21 users click on an ad (𝑝̂ =
"%
#$
!"
#$
= 42%). Format B had 19 users click on an ad (𝑝̂ =
= 38%).
(1) Calculate the difference (A – B) in proportion of users who clicked on an ad.
(2) In your own words, explain what ‘statistically significant’ means. Before doing further calculations,
do you believe the difference you calculated in part (a) is statistically significant? Explain.
The following simulation was performed assuming format A and format B are equally effective: the
individual outcomes from this experiment (‘click’ or ‘not clicked’) were randomly sorted into two
groups (representing formats A and B). The proportion who ‘clicked’ were calculated in each group
and then the difference was taken between the two groups’ proportions. This was repeated 100
times. The resulting differences in proportions who clicked are displayed here:
(3) Using the results from the simulation, do you believe the difference in proportions of users who
clicked between formats A and B is statistically significant? Explain.
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
183
Does SAT prep improve scores? Part 2.
Some students at your school claim that taking an SAT prep course will help improve their SAT
score. Design a study to help discover if this claim is true.
Here are six proposed studies for investigating the question of the day. Suppose we found that
the mean SAT score of students who took an SAT prep course is significantly higher than the
mean SAT score of students who didn’t take an SAT prep course. What conclusions could we
make? Can we generalize and can we determine causation?
1. Get all the students in a senior-only class to participate in a study. Ask them whether or not they took
an SAT prep course and divide them into two groups based on their answer to this question.
Conclusion:
2. Select a random sample of Seniors from your school to participate in a study. Ask them whether or not
they took an SAT prep course and divide them into two groups based on their answer to this question.
Conclusion:
3. Select a random sample of students who have taken the SAT from your school to participate in a study.
Ask them whether or not they took an SAT prep course and divide them into two groups based on their
answer to this question.
Conclusion:
4. Get all the students in a senior-only class to participate in a study. Randomly assign half of the
students to take an SAT prep course and have the remaining half not take an SAT prep class. Then have
both groups take the SAT again.
Conclusion:
5. Select a random sample of Seniors from your school to participate in a study. Randomly assign half of
the students to take an SAT prep course and have the remaining half not take an SAT prep course. Then
have both groups take the SAT again.
Conclusion:
6. Select a random sample of students who have taken the SAT from your school to participate in a study.
Randomly assign half of the students to take an SAT prep course and have the remaining half not take an
SAT prep course. Then have both groups take the SAT again.
Conclusion:
184
Scope of Inference
Important Ideas:
Check Your Understanding:
1. Zach works at the Verizon store and wonders if iPhones last longer if the screen brightness is
set to low. He selects a random sample of 20 brand new iPhones from this store and
randomly splits them into two groups of 10. For the first group of 10 iPhones, he sets the
screen brightness to low and then starts a movie. For the second group of 10 iPhones, he
sets the screen brightness to high and then starts a movie. For each iPhone, he measures
the amount of time until the battery is all the way dead. He finds that the low brightness
iPhones lasted longer, on average, than the high brightness iPhones.
(a) Was this an observational study or an experiment? Explain your reasoning.
(b) What is the explanatory variable and the response variable?
(c) Was a random sample used to collect the data?
(d) Was random assignment used to set up an experiment?
(e) What conclusion can we make from this study?
185
Does caffeine increase heart rate? The Finale.
Today we will be testing to see if caffeine increases heart rate.
1. Begin by taking your heart rate (beats per minute).
2. Mrs. Gallas took a random sample of 5 students’ heart rates. The sample mean
was 64.5 beats per minute with a margin of error of 5.6 beats. Can she expect
that the sample mean is the same as the population mean? Explain.
3. Would you be surprised to learn that the true population mean was 68 beats per
minute? Explain.
4. How could Mrs. Gallas find a sample mean that is closer to the true population?
5. Mrs. Gallas will give you a beverage to drink. Drink the entire beverage. Is this
an observational study or an experiment? Explain the difference.
6. Was blinding used for this experiment? Why is that important?
7. Describe what a confounding variable is and list a few that might affect our
results.
8. If we really just want to see if caffeine increase heart rate, why can’t we just have
one group who all drinks caffeinated coke?
186
9. Identify each of the following:
Explanatory variable:
Response variable:
Experimental units:
Treatments:
10. Describe how we could randomly assign the students in the class into two groups
(Caffeine and Caffeine-free).
11. Why do we use random assignment to create the two groups? What is its
purpose?
12. Draw an outline of a completely randomized design.
13. What are the four key principles of experimental design? Explain how your
experiment designed in #12 meets all of them.
14. Mrs. Gallas thinks that gender may play a role in the effect that caffeine has on
heart rate. How could we adjust our experimental design to account for that?
Explain.
15. Outline a randomized block design.
187
16. Could we use a matched pairs design to test the effects of caffeine? If yes,
describe the process of pairing. If no, explain why not.
17. We will be using the completely randomized design. It’s time to take your heart
rate again. What is your heart rate now?
How much did your heart rate
increase? (New – Initial)
Add your change in heart rate to the table on
the board.
Increase in heart rate
Caffeine
Caffeinefree
18. Go to stapplet.com and click One Quantitative Variable. Change the Number of
Groups to 2 and add the data for both groups. Record the mean of each and find
the difference.
Mean with Caffeine:___________
Mean with No Caffeine:__________
Difference (Caffeine – No Caffeine):____________
19. Under Perform Inference, choose “Simulate difference in two means” from the
drop down menu. Add 50 samples. Sketch the dot plot below.
20. What percent of the dots are larger than the difference in means we got from our
experiment? Is the difference statistically significant? Explain.
188
21. Can we generalize the results? Explain.
22. Can we determine cause and effect? Explain.
23. What conclusion can we make?
24. Was this experiment conducted in an ethical manner? Explain.
189
2. Researchers are investigating the effectiveness of using a fungus to control the spread of an insect that destroys
trees. The researchers will create four different concentrations of fungus mixtures: 0 milliliters per liter (ml/L),
1.25 ml/L, 2.5 ml/L, and 3.75 ml/L. An equal number of the insects will be placed into 20 individual containers.
The group of insects in each container will be sprayed with one of the four mixtures, and the researchers will
record the number of insects that are still alive in each container one week after spraying.
(a) Identify the treatments, experimental units, and response variable of the experiment.
Treatments:
Experimental units:
Response variable:
(b) Does the experiment have a control group? Explain your answer.
(c) Describe how the treatments can be randomly assigned to the experimental units so that each treatment
has the same number of units.
190
®
2014 AP STATISTICS FREE-RESPONSE QUESTIONS
4. As part of its twenty-fifth reunion celebration, the class of 1988 (students who graduated in 1988) at a state
university held a reception on campus. In an informal survey, the director of alumni development asked 50 of
the attendees about their incomes. The director computed the mean income of the 50 attendees to be $189,952.
In a news release, the director announced, “The members of our class of 1988 enjoyed resounding success.
Last year’s mean income of its members was $189,952!”
(a) What would be a statistical advantage of using the median of the reported incomes, rather than the mean,
as the estimate of the typical income?
(b) The director felt the members who attended the reception may be different from the class as a whole.
A more detailed survey of the class was planned to find a better estimate of the income as well as other
facts about the alumni. The staff developed two methods based on the available funds to carry out the
survey.
Method 1: Send out an e-mail to all 6,826 members of the class asking them to complete an online form.
The staff estimates that at least 600 members will respond.
Method 2: Select a simple random sample of members of the class and contact the selected members
directly by phone. Follow up to ensure that all responses are obtained. Because method 2 will
require more time than method 1, the staff estimates that only 100 members of the class could be
contacted using method 2.
Which of the two methods would you select for estimating the average yearly income of all 6,826 members
of the class of 1988 ? Explain your reasoning by comparing the two methods and the effect of each method
on the estimate.
191
192
Unit 4
Chapter 5
Probability
193
AP Statistics Handout: Lesson 5.1
Topics: law of large numbers, definition and properties of probability, probability models
Lesson 5.1 Guided Notes
Probability and Statistics
estimate
Probabilities
Statistics
generate
The Law of Large Numbers
Law of large numbers: After many ______________, the relative frequency of outcomes will approach
their ________________________.
Coin flips applet: http://digfir-published.macmillanusa.com/stats_applet/stats_applet_10_prob.html
-Source: Digital First project from Bedford, Freeman, & Worth publishers
In your own words, describe how this
simulation demonstrates the law of large
numbers:
The “due for one” myth
You have a fair coin. You flip it 10 times. In which situation is getting tails (T) on your next flip more
likely?
Situation 1: HTTHHTHTHT
Situation 2: HTTHHHHHHH
194
A tale of two Khris/Chris Davis’s
a) How many at-bats did Chris Davis have in a row without
recording a hit?
b) What does Chris Davis’s story tell us about the “due for one”
myth?
Chris Davis
c) Honestly, the Khris Davis batting average streak isn’t really related
to the “due for one” myth. However, because Khris Davis’s streak is
so incredible and because his name is so similar to the other Davis, I
just HAD to include it. So, there’s no question here that requires a
response. I hope you enjoyed this part of the lesson!
Khris Davis
The Simple Definition of Probability
Probability =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑎𝑛 𝑒𝑣𝑒𝑛𝑡
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
The above equation is the simple definition of probability
• Only used when all outcomes are _________________________
Example: There are 8 red marbles and 12 blue marbles in a jar. What is the probability of selecting a red
marble from the jar? Let R = the event of selecting a red marble. P(R) = probability of event R occurring.
Properties of Probabilities
1. Probabilities are always between _________ (0% - 100%).
2. The probabilities of all possible events ________________ (100%).
3. Complement rule: the probability of event A ______ happening, P(AC), is equal to 1 – P(A).
195
Probability Models
Probability models: List a set of _________________ and their probabilities.
Properties:
1. All probabilities are between 0 – 1.
2. All probabilities ____________________.
Example: I have a bag of 20 beads. They are a mix of clear/solid and red/green. There are 6 clear red
beads in the bag and 8 red solid beads. There are 2 clear green beads and 4 green solid beads.
a) Fill out the probability model
Bead Type
clear red
red solid
clear green
green solid
Probability
b) Verify that this is a valid probability model (check the probability model properties):
Let: R = Select a red bead, Q = Select a red solid bead
c) Find these probabilities, showing your work:
1. P(Q) =
2. P(R) =
3. P(QC ) =
4. P(RC ) =
Lesson 5.1 Discussion
2011 NYC “Stop and Frisk” police stops:
Race-Ethnicity
Black
Hispanic
White
Proportion of Stops
55.1%
35.2%
9.7%
Proportions calculated just among these three groups (the predominant groups in the dataset)
Data source: NYC.GOV, https://www1.nyc.gov/site/nypd/stats/reports-analysis/stopfrisk.page
Population of NYC in 2011:
Race-Ethnicity
Black
Hispanic
White
Population
2,054,101
2,373,304
2,731,173
Data source: 2011 American Community Survey (data.census.gov)
196
Discussion Question: If stops were done
completely at random, which groups would be
stopped proportionally more often? Less often?
What conclusions can you draw from this?
How good is Mrs. Gallas at free throws?
Mrs. Gallas thinks she is a pretty good free throw shooter. How many free throws would you
like to see Mrs. Gallas shoot before you could be confident guessing her free throw
percentage? We’ll watch Mrs. Gallas shoot free throws, when you are confident make a guess
at her free throw percentage.
1. As each shot is attempted, keep track of the number of made free throws and the total
number of shots attempted in the table below. When you think you know Mrs. Gallas’
true free throw percentage, stop recording the shots.
Shot #
1
2
3
4
5
10
15
20
30
40
50
60
70
Result
(Make or Miss)
Proportion of
Makes
2. What do you think Mrs. Gallas’ true free throw percentage is?
3. Sketch the graph displaying the proportion of made free throws.
4. How could you make your guess more accurate?
5. Mrs. Gallas has a ____% probability of making a free throw. Interpret this probability.
197
80
The Idea of Probability
Important ideas:
Check Your Understanding
1. Use 4 of the following probabilities to complete the middle column of the table, then give an
example of an event may have this probability. Probabilities: 0, 0.001, 0.3, 0.6, 0.99, 1
Explanation
This outcome is impossible. It can never
occur.
This outcome is certain. It will occur on every
trial.
This outcome is very unlikely, but it will occur
once in a while in a long sequence of trials.
This outcome will occur more often than not,
but doesn’t occur almost every time.
Probability
Example
2. In the Wheel of Fortune, there is a 1/9 = 11.1% probability of spinning “Bankrupt” on any given
spin. Interpret the probability.
3. In 9 consecutive spins of the Wheel of Fortune, none of the spins land on “Bankrupt”. The next
contestant worries that he is more likely to land on Bankrupt now because Bankrupt is due.
Explain why this thinking is wrong.
198
Are Soda Contests True?
Pepsi ran a promo contest for their 20 oz. bottles of soda. Some of the caps said, “Please try
again!” while others said, “You’re a winner!” Pepsi advertised the promotion with the slogan “1
in 6 wins a prize.” Mrs. Gallas’ statistics class wonders if the company’s claim is true. To find
out, all 30 students in the class go to the store, and each buys one 20-ounce bottle of the
soda. Two of the 30 students get caps that say “You’re a winner!”
1. How many winners would you expect to get out of a class of 30? Is it guaranteed?
Does this result give convincing evidence that the company’s 1-in-6 claim is inaccurate? We
will perform a simulation to help answer this question. We will assume Pepsi is telling the
truth. If they are telling the truth, what is the probability of getting 2 or fewer winners in a class
of 30 purely by chance? Let’s find out.
2. What could we use to model a 1/6 probability?
to “Losers” and “Winners”. List them below.
Assign certain outcomes
3. Roll your die 30 times to imitate the process of the students in Mrs. Gallas’ statistics
class buying their sodas. How many of them won a prize?
4. Repeat steps 1 and 2. How many won a prize this time?
5. Plot the number of prize winners for each trial of 30 to the dot plot on the board. (2 dots)
6. Sketch the class dot plot below.
7. What percent of the time did Mrs. Gallas’ statistics class get two or fewer prizes, just by
chance?
8. Does it seem plausible that the company is telling the truth but that the class just got
unlucky? Or do we have convincing evidence that Pepsi is lying?
199
Simulation
Important ideas:
Check Your Understanding
Some people have wondered if the Wheel of Fortune wheel is fair. If it is
fair, the “Bankrupt” result should occur with probability 1/9. To test the
wheel, Pat Sajak observed a random sample of 50 spins of the wheel and
found that “Bankrupt” was spun 10 times. Is this convincing evidence that
Bankrupt comes up more often than it should?
a. Describe how you would carry out a simulation to estimate the
probability that in a fair wheel, Bankrupt comes up 10 or more times.
The dotplot displays the results of 100
simulated trials of 50 spins of a fair wheel.
b. Explain what the one dot above 12 indicates.
c. What conclusion would you draw about 10 or more Bankrupt spins occurring in 50 spins?
Explain your answer.
200
5.1 PRACTICE!
The Golden Ticket
At a local high school, 95 students have permission to park on campus. Each month, the student council holds a “golden
ticket parking lottery” at a school assembly. The two lucky winners are given reserved parking spots next to the school’s
main entrance. Last month, the winning tickets were drawn by a student council member from the AP Statistics class.
When both golden tickets went to members of that same AP Stats class, some people thought the lottery had been
rigged. There are 28 students in the AP Statistics class, all of whom are eligible to park on campus. Design and carry out
a simulation using the Table of Random Digits (beginning with line 139) to decide whether it’s plausible that the lottery
was carried out fairly.
Do:
Trial #
Numbers
Selected
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Both in AP
Stats?
Trial #
Numbers
Selected
Both in AP
Stats?
Conclude:
201
NASCAR
In an attempt to increase sales, Cheerios decides to offer a NASCAR promotion. Each box of cereal will contain a
collectible card featuring one of these NASCAR drivers: Jeff Gordon, Dale Earnhardt, Jr., Tony Stewart, Danica Patrick, or
Jimmie Johnson.
The company says that each of the 5 cards is equally likely to appear in any box of cereal. A NASCAR fan decides to keep
buying boxes of the cereal until she has all 5 drivers’ cards. She is surprised when it takes her 23 boxes to get the full set
of cards. Should she be surprised? Design and carry out a simulation using your graphing calculator to help answer this
question.
Trial #
Numbers
Selected
Total # of
Boxes
Trial #
Numbers
Selected
Total # of
Boxes
1
2
3
4
5
6
7
8
9
10
Trial #
Numbers
Selected
Total # of
Boxes
Trial #
Numbers
Selected
11
12
13
14
15
16
17
18
19
20
Total # of
Boxes
Conclude:
202
AP Statistics Handout: Lesson 5.2
Topics: unions & intersections, the addition rule in multiple formats, mutually exclusive events
Lesson 5.2 Guided Notes
Unions & Intersections
For each of the following probability expressions, take notes on what the notation means and shade the
Venn Diagram:
1. P(A):
3. P(AC):
2. P(B):
A
B
A
4. P(A∩B):
B
A
6. P((A∪B)C):
5. P(A∪B):
A
B
B
A
B
A
B
The Addition Rule in Probability Models
The OKCupid Dataset - https://github.com/wetchler/okcupid
This dataset is composed of public online dating profiles from 2012 on OKCupid. We filtered for people
who identified as age 36, male, and heterosexual, living in the San Francisco area. After filtering, we
collected data for each profile that reported their height and yearly earnings.
We’ll categorize the 192 men in the OKCupid dataset using the following convention:
Shorter than median
Taller than median
Earns less than median ($)
Earns more than median ($)
Short Low-Earner
Short High-Earner
Tall Low-Earner
Tall High-Earner
Socially de-valued
Socially valued
Estimated 2012 median yearly earnings for individual men in San Francisco county: $59,397. Male median height (U.S.): 69.2 in. (5’9”). 1. Earnings
median is an upper-bound estimate based on median earnings for workers in San Francisco county per the 2012 American Community Survey
(data.census.gov). 2. Height median is from the CDC: National Health Statistics Reports, “Mean Body Weight, Height, Waist Circumference, and Body
Mass Index Among Adults: United States, 1999–2000 Through 2015–2016.” Center for Disease Control and Prevention, 2018.
203
In the OkCupid data, we find 21 short low-earners, 22 tall low-earners, 48 short high-earners, and 101
tall high-earners.
Category
short low-earner
tall low-earner
short high-earner
tall high-earner
Probability
21
= 0.109
192
22
= 0.115
192
48
= 0.250
20
101
= 0.526
192
a) Check that this is a valid probability model:
Let: T = event of selecting a tall man
H = event of selecting a high-income earner
1. P(T) =
2. P(T∩H) =
3. P(T∪H) =
Simple Addition Rule: “Or” means ________
• The ___________ P(A∪B) brings together A, B, and A∩B. So
you add the probabilities together.
• Think unions → _______________________
A
B
The Addition Rule in Two-Way Tables
low-earner
high-earner
total
short
21
48
69
tall
22
101
123
total
43
149
192
Formal Addition Rule: P(A∪B) = P(A) + P(B) – P(A∩B)
204
1. P(T∩HC) =
2. P(H) =
3. P(T∪H) =
_______________ Addition Rule:
• “Or” means add
• Avoid _________________________
The Addition Rule in Venn Diagrams
T
H
22
101
48
low-earner
high-earner
total
short
21
48
69
tall
22
101
123
total
43
149
192
21
1. P(H) =
2. P(H∩T) =
4. P(HC) =
5. P(T∪H) =
3. P(H∩TC) =
Mutually Exclusive Events
Mutually exclusive: when events that have no __________________ (i.e. they cannot _________ occur)
A
B
For mutually exclusive events only:
P(A∪B) = P(A) + P(B)
•
Don’t have to worry about _____________________
Lesson 5.2 Discussion
Discussion Question: Do you believe many of the men on
OKCupid are lying on their profiles? Provide a statistical
explanation for your answer.
205
AP Statistics Handout: Lesson 5.3
Topics: conditional probability, independence, the multiplication rule, tree diagrams
Lesson 5.3 Guided Notes
Conditional Probability
Condition: a “____________” in a problem
• P(A|B) = Probability of A occurring given that B _____________________________
The OKCupid Dataset - https://github.com/wetchler/okcupid
This dataset is composed of public online dating profiles from 2012 on OKCupid. We filtered for people
who identified as age 36, male, and heterosexual, living in the San Francisco area. After filtering, we
collected data for each profile that reported their height and yearly earnings.
We’ll categorize the 192 men in the OKCupid dataset using the following convention:
Shorter than median
Taller than median
Earns less than median ($)
Earns more than median ($)
Short Low-Earner
Short High-Earner
Tall Low-Earner
Tall High-Earner
Socially de-valued
Socially valued
Estimated 2012 median yearly earnings for individual men in San Francisco county: $59,397. Male median height (U.S.): 69.2 in. (5’9”). 1. Earnings
median is an upper-bound estimate based on median earnings for workers in San Francisco county per the 2012 American Community Survey
(data.census.gov). 2. Height median is from the CDC: National Health Statistics Reports, “Mean Body Weight, Height, Waist Circumference, and Body
Mass Index Among Adults: United States, 1999–2000 Through 2015–2016.” Center for Disease Control and Prevention, 2018.
In the OkCupid data, we find 21 short low-earners, 22 tall low-earners, 48 short high-earners, and 101
tall high-earners. Let: T = event of selecting a tall man, H = event of selecting a high-income earner.
T
H
22
101
1. P(T) =
48
2. P(T|H) =
21
Conditional Probability Formula:
P(A|B) =
𝑃(𝐴∩𝐵)
𝑃(𝐵)
Conditional Probability Intuition:
• “Given” means ____________ by the given
206
Conditional Probability in Two-Way Tables
low-earner
high-earner
total
short
21
48
69
tall
22
101
123
total
43
149
192
1. P(T) =
2. P(T|HC) =
Independence
Independence: Two events (A & B) are independent if knowing the outcome of one event __________
_______ affect the probability that the other event will occur.
•
P(A|B) = _______
Example: Are the events “selecting a tall man” and “selecting a man who is a low-earner” independent?
Use the two probabilities you calculated for the two-way table above to support your answer.
Answer:
When we knew the person was a low-earner (when HC was “given”), the probability of selecting a tall
person ___________. So, the events of selecting a low earner and someone who is tall are ____
___________________.
In other words, their earnings ______________________________ about their height. So, earnings and
height are not independent.
Independent vs. Mutually Exclusive
B
A
0.80
0.20
Mutually exclusive: when events that have no intersection
(i.e. they cannot both occur)
P(A) =
P(A|B) =
Mutually exclusive events are _____________________________. Knowing that one event occurs
greatly affects the probability of the other event (_____________________________).
207
Multiplication Rule
Formal Addition Rule:
For all events: P(A∩B) = P(A) × P(B|A)
_______________ Multiplication Rule:
• “And” means _______________
• Account for _________________ events
For independent events: P(A∩B) = P(A) × P(B)
Tree Diagrams
Tree diagrams: Models multiple __________________ or successive events (events that depend on each
other or that happen right after each other).
• Each ______________ represents a different event
Dream School Example:
1. You have a 0.65 probability of getting higher than average GPA and ACT scores.
2. If you have a higher than average GPA/ACT, you have a 0.83 chance of being admitted.
3. If you have a below average GPA/ACT score, you have a 0.39 chance of being admitted.
Draw the tree diagram here:
Let:
H = Event of getting higher than average ACT/GPA , A = Event of being admitted to your dream school
1. P(H∩A) =
2. P(A) =
3. P(H|A) =
208
5 Intuitive Probability Rules for a 5
1. Probabilities are between 0 – 1 (inclusive)
2. Complement rule: P(AC) = 1 – P(A)
3. “Or” means add, beware of double-counts
4. “Given” means divide by the given
5. “And” means multiply, adjust for dependence
If you depend on formulas
Lesson 5.3 Discussion
low-earner
high-earner
total
short
21
48
69
tall
22
101
total
43
149
low-earner
high-earner
total
short
48.8%
48
69
123
tall
51.2%
101
123
192
total
100%
149
192
2. P(T|HC) = 51.2%
1. P(T) = 64.1%
If we find the conditional distribution (“given” low-earner), we’re close to the 50-50 split we’d expect
around the median height!
Discussion Question: Do you believe that filtering results for just the men who reported below-median
earnings would return more honest matches? Explain your reasoning.
209
Who will win the Last Banana?
Suppose that you’re on a desert island playing dice with another castaway. The winner’s prize
will be the last banana. Here are the rules of the game:
• Each player rolls a die
• If the largest value shown is a 1, 2, 3, or 4, then Player 1 wins
• If the largest value shown is a 5 or 6 then Player 2 wins
1. Who do you think has advantage in this game: Player 1, Player 2, or neither? Make your best
guess and explain your choice.
2. Play the game 20 times with your partner and record the winner of each game by tallying in the
table below.
Player
1
2
Tally/Count of Wins
Percentage of Wins
a. How many times did Player 1 win?
Write this as a proportion.
b. How many times did Player 2 win?
Write this as a proportion.
3. Who won more often? Maybe this was only true for your group. Let’s see how the rest of the class
did. Write the number of wins for Player 1 in the table on the board.
a. Find the total proportion of wins for Player 1 for the whole class.
b. Find the total proportion of wins for Player 2 for the whole class.
4. To determine the true probability of Player 1 winning,
we should list out all possible rolls that we could get.
Complete the table below to show all possible rolls.
a. Use your table to find the probability of
Player 1 winning.
1
1
2
3
4
b. Which was closer to the probability you
found in #4a, your group data or the
classroom data? Why do you think that is?
210
5
6
1,1
2
3
4
5
6
Basic Probability Rules
Important ideas:
Check Your Understanding
Pew Research Center reported that among mothers, family size is shrinking. Suppose we are to
randomly select one mother (age 40 to 44) and record the number of children she has. Here is a
probability model.
Number of Children
Probability
1
0.22
2
0.41
3
0.24
4+
0.13
(a) Explain why this is a valid probability model.
(b) Explain why events “have 1 child” and “have 2 children” are mutually exclusive
For each of the following write the event using proper notation and find the probability:
(c) Find the probability that a randomly selected mother has less than 2 children.
(d) Find the probability that a randomly selected mother has 1 or 2 children.
(e) Find the probability that a randomly selected mother does not have 4 or more children.
211
Can You Taco Tongue and Evil Eyebrow?
Some people believe that the ability to taco tongue and evil eyebrow is something that you
are born with. Is this true? Are the two abilities somehow related?
1. Collect class data to fill in the following two-way table and Venn Diagram.
Yes
Evil Eyebrow
No
Evil Eyebrow
Total
Yes Taco Tongue
____
_
No Taco Tongue
Yes
Evil Eyebrow
Yes
Taco Tongue
Total
_____
_____
_____
2. Suppose that we randomly choose a student from class. Find the following probabilities.
P(Yes Taco Tongue) =
P(Yes Evil Eyebrow) =
P(No Taco Tongue) =
P(No Evil Eyebrow) =
P(Yes Taco Tongue AND Yes Evil Eyebrow) =
P(Yes Evil Eyebrow AND No Taco Tongue) =
P(Yes Taco Tongue AND No Evil Eyebrow) =
P(No Taco Tongue AND No Evil Eyebrow) =
3. Suppose that we randomly choose a student from class. Find the following probabilities.
P(Yes Evil Eyebrow) =
P(No Evil Eyebrow) =
P(Yes Evil Eyebrow OR No Evil Eyebrow) =
4. Suppose that we randomly choose a student from class. Find the following probabilities.
P(Yes Taco Tongue) =
P(Yes Evil Eyebrow) =
P(Yes Taco Tongue OR Yes Evil Eyebrow) =
212
Venn Diagrams, General Addition Rule
Important Ideas:
Check Your Understanding:
A random sample of high school students were surveyed regarding their toilet paper habits. Should
the toilet paper pull over the roll? Do they replace the roll when it is empty? The two-way table
displays the data. Suppose we choose a member of the sample at random. Define events O: Over
and R: Replace.
Prefer the toilet paper to pull over the roll
Yes
No
Replaces the roll
Yes
58
24
when it is empty
No
12
22
a. Explain in plain language what P(OC) means and find the probability.
b. Explain why P(O or R) ≠ P(O) + P(R). Then find P(O or R).
c. Make a Venn diagram to display the sample space of this chance process.
d. Write the event “Does not replace the roll and prefers the toilet paper pulls over the roll” in
symbolic form and find the probability.
213
5.2 PRACTICE!
Languages in Canada
Canada has two official languages, English and French. Choose a Canadian at random and ask, “What is your mother
tongue?” Here is the distribution of responses, combining many separate languages from the broad Asia/Pacific region:
(a) What probability should replace “?” in the distribution? Why?
(b) What is the probability that a Canadian’s mother tongue is not English?
(c) What is the probability that a Canadian’s mother tongue is a language other than English or French?
U.S. Senators
The two-way table below describes the members of the U.S Senate in a recent year.
(a) Who are the individuals? What variables are being measured?
(b) If we select a U.S. senator at random, what’s the probability that we choose
i.
a Democrat?
ii. a female?
iii. a female Democrat?
iv. a female or a Democrat?
214
Phone Usage
According to the National Center for Health Statistics, in December 2008, 78% of US households had a traditional
landline telephone, 80% of households had cell phones, and 60% had both. Suppose we randomly selected a household
in December 2008.
We will define events A: has a landline and B: has a cell phone.
Landline
No Landline
Total
Cell Phone No Cell Phone Total
0.60
0.18
0.78
0.20
0.02
0.22
0.80
0.20
1.00
(a) Construct a Venn diagram to represent the outcomes of this chance process.
(b) Find the probability that the household has at least one of the two types of phones.
(c) Find the probability the household has a cell phone only.
215
Can You Taco Tongue and Evil Eyebrow? Day 2
Are the events “Yes Taco Tongue” and “Yes Evil Eyebrow” independent?
1. Find class data from the previous lesson and fill in the following two-way table.
Yes
Evil Eyebrow
No
Evil Eyebrow
Total
Yes Taco Tongue
No Taco Tongue
Total
Suppose we randomly choose a student from class.
2. Find P(Yes Taco Tongue OR Yes Evil Eyebrow)
3. Given that the person selected is a Yes Evil Eyebrow, what is the probability that they are
a Yes Taco Tongue? Write as a fraction, a decimal, and a percent.
4. Given that the person selected is a No Evil Eyebrow, what is the probability that they are
a Yes Taco Tongue? Write as a fraction, a decimal, and a percent.
Definition: Two events are independent if knowing whether or not one event has occurred
does not change the probability that the other event will occur.
5. Are the events “Yes Taco Tongue” and “Yes Evil Eyebrow” independent? Explain.
216
Consider the data for all Seniors at EKHS.
Yes
Evil Eyebrow
No
Evil Eyebrow
Total
Yes Taco Tongue
180
300
480
No Taco Tongue
20
100
120
Total
200
400
600
6. Find each of the following using the data in the table. Write as a fraction, a decimal, and a
percent.
a. P(Yes Taco Tongue) =
b. P(Yes Taco Tongue | Yes Evil Eyebrow) =
c. P(Yes Taco Tongue | No Evil Eyebrow) =
d. Are “Yes Taco Tongue” and “Yes Evil Eyebrow” independent?
7. Fill in the table as if the events were INDEPENDENT.
Yes
Evil Eyebrow
No
Evil Eyebrow
Total
Yes Taco Tongue
480
No Taco Tongue
120
Total
200
400
600
8. Find each of the following using the INDEPENDENT table. Write as a fraction, a decimal,
and a percent.
a. P(Yes Taco Tongue) =
b. P(Yes Taco Tongue | Yes Evil Eyebrow) =
c. P(Yes Taco Tongue | No Evil Eyebrow) =
9. What do you notice about your answers in #6 and #8?
10. Generalize: Complete the following statement using a formula.
Let A à Yes Taco Tongue and B à Yes Evil Eyebrow
If events A and B are INDEPENDENT then…
217
Conditional Probability and Independence
Important Ideas:
Check Your Understanding:
The Pew Research center randomly selected 100 mothers age 40 to 44 in 1976, 1994, and 2014 and
asked each mother how many children they have. The two-way table summarizes the responses.
Number
of
children
1 child
2 children
3 children
4+ children
Total
1976
11
24
25
40
100
Year
1994
21
43
23
13
100
2014
22
41
24
13
100
Total
54
108
72
66
300
Suppose we randomly select one of the survey respondents. Define events C: have 4 or more
children, S: 1976, N: 1994, and F: 2014.
1. Find P(C | S). Interpret this value in context.
2. Given that the chosen mother was not surveyed in 1976, what’s the probability that she has 4 or
more children? Write your answer as a probability statement using correct symbols for the events.
3. Are the events “Surveyed in 1976” and “4 or more children” independent? Explain.
218
AP Statistics Handout: Lesson 5.4
Topics: successive independent events, “at least one,” sampling w/o replacement
Lesson 5.4 Guided Notes
Successive Independent Events
Successive: Events are successive if one happens after the other (i.e. ____________________________)
Independent: Events are independent if knowing the outcome of one event _______________________
the probability of another event.
Example 1: What is the probability of getting “heads” 5x in a row?
Since the probabilities _______________________, you don’t need a tree diagram!
Example 2: A free throw is a penalty shot in basketball. One of the NBA’s best free throw shooters is
Stephen Curry. He makes 90.6% of the free throws he attempts. Assume each free throw attempt is
independent. What is the probability he makes 6 free throws in a row?
Example 3: Stephen Curry is fouled in many 3-point situations, so he often takes 3
free throws at a time. When he takes 3 free throws, what is the probability he makes
the first two but misses the last one?
219
“At Least One” Scenarios
Example: Stephen Curry makes 90.6% of the free throws he attempts. Assume each of his free throw
attempts are independent. What is the probability he makes at least one of his next 4 free throws?
“At least one” scenarios: When asked to find the probability of “at least one” occurrence of a successive
independent event, take the ______________________ of the event that ___________ occur.
P(at least one) = ______________________
Sampling Without Replacement
Example: You have a jar with 12 blue marbles and 8 red marbles. Imagine you sample marbles without
replacement. What is the probability of drawing the following: blue, then red, then blue.
Lesson 5.4 Discussion
Testing Kobe’s “Hot Hand”
The “hot hand” theory: When a player starts to make many shots in a row, the have a “hot hand” – their
probability of making shots is higher than normal.
• Your shots are _________________________. Your
current shot probability ________________ on your
previous shots.
a) Let’s assume the “hot hand” doesn’t exist. Each shot has an
equal and independent probability of going in (44.7%). What is
the probability of Kobe making an insane 8 shots in a row?
Kobe’s 3rd quarter during his 81-point game
b) In his career, Kobe took 30,697 shots. Given this information and the probability of an 8-shot streak
(calculated above), what is the expected number of times Kobe would get an 8-shot streak by chance?
220
Can you get a pair of Aces or a pair of Kings?
Rules of the game. Five cards total: two aces and three Kings. The player chooses their
first card and records the results, and then chooses their second card (without replacement)
and records the result. The player wins if they get a pair of Aces or a pair of Kings.
1. Choose one person who is the dealer and one who is the player. Play the game 10 times.
First card
Second card
Winner?
Based on your 10 games, what is the probability of winning this game? __________
2. Go to the front of room to record the number of wins in 10 games.
Based on the whole class data, what is the probability of winning this game? __________
3. Let’s try to use a Tree Diagram to calculate the theoretical probability. Fill in the blank boxes
with the correct probabilities.
st
1 card
is Ace
2nd card
is Ace
= P(1st is Ace AND 2nd is Ace)
2nd card
is King
= P(1st is Ace AND 2nd is King)
2nd card
is Ace
= P(1st is King AND 2nd is Ace)
2nd card
is King
= P(1st is King AND 2nd is King)
Ace Game
1st card
is King
4. Find the theoretical probability of winning the game. ____________________________
5. What is the probability that the 1st card was a King, given that the person won the game?
221
Conditional Probability and Independence
Important Ideas:
Check Your Understanding:
In the 2016 election, 30 states went to the Republican candidate and 20 states went to the
Democratic candidate. Of the 30 states that went Republican, 29 were in the continental United
States. Of the 20 states that went Democratic, 19 are in the continental United States. One state is
selected at random.
a. Construct a tree diagram to model this chance process.
b. Find the probability that a randomly selected state is in the continental U.S. and went Republican.
c. If we select 4 states at random (with replacement) what is the probability that at least 1 of the
states is in the continental U.S. and went Republican?
d. Given that a randomly selected state is not in the continental U.S., what is the probability that it
went Republican?
222
5.3 PRACTICE!
The Probability of a Flush
1. A poker player holds a flush when all 5 cards in the hand belong to the same suit. Remember that a deck contains 52
cards, 4 suits (hearts, diamonds, spades and clubs) with 13 cards of each suit. When the deck is well shuffled, each card
dealt is equally likely to be any of those that remain in the deck.
(a) We will concentrate on spades. What is the probability that the first card dealt is a spade? What is the conditional
probability that the second card is a spade given that the first is a spade (partial tree diagram may help)?
(b) Continue to count the remaining cards to find the conditional probabilities of a spade on the third, the fourth, and
the fifth card given in each case that all previous cards are spades (consider continuing with your tree diagram from
above).
(c) The probability of being dealt 5 spades is the product of the five probabilities you have found. Why? What is this
probability?
(d) The probability of being dealt 5 hearts or 5 diamonds or 5 clubs is the same as the probability of being dealt 5 spades.
What is the probability of being dealt a flush?
223
2. Internet sites often vanish or move, so that references to them can’t be followed. In fact, 13% of Internet sites
referenced in major scientific journals are lost within two years after publication. If a paper contains seven Internet
references, what is the probability that at least one of them doesn’t work two years later? Show your work!
3. A recent census at Emory University revealed that 40% of its students mainly used Macintosh computers (Macs). The
rest mainly used PCs. At the time of the census, 67% of the school’s students were undergraduates. The rest were
graduate students. In the census, 23% of the respondents were graduate students who said that they used PCs as their
primary computers. Suppose we select a student at random from among those who were part of the census and learn
that the student mainly uses a Mac. Find the probability that this person is a graduate student. Show your work... (Hint:
a tree diagram may be too difficult to construct, so consider a two-way table)
4. Many employers require prospective employees to take a drug test. A positive result on this test indicates that the
prospective employee uses illegal drugs. However, not all people who test positive actually use drugs. Suppose that 4%
of prospective employees use drugs, the false positive rate is 5% and the false negative rate is 10%. What percent of
people who test positive actually use illegal drugs?
(Hint: look at the rule on p. 324 first, then study a similar example on p. 326 ... this is tough, but you can do this!!)
224
5.3 PRACTICE! #2
Picking Two Sneezers
This is a two-way table that classified 40 students according to their gender and whether they had allergies.
Allergies
No Allergies
Total
Female Male Total
10
8
18
13
9
22
23
17
40
Suppose we chose 2 students at random.
a. Draw a tree diagram or chart that shows the sample space for this chance process.
b. Find the probability that both students selected suffer from allergies.
c. Find the probability that neither student selected suffers from allergies.
d. Find the probability that at least one student selected suffers from allergies.
e. Find the probability that only one student selected suffers from allergies.
225
Media Usage and Good Grades
In January 2010, the Kaiser Family Foundation released a study about the influence of media in the lives of young people
ages 8-18 (http://www.kff.org/entmedia/mh012010pkg.cfm). In the study, 17% of the youth were classified as light
media users, 62% were classified as moderate media users and 21% were classified as heavy media users. Of the light
users who responded, 74% described their grades as good (A’s and B’s), while only 68% of the moderate users and 52%
of the heavy users described their grades as good.
a. According to this study, what percent of young people ages 8-18 described their grades as good? Use a tree
diagram or chart to calculate the probability.
b. According to the tree diagram you constructed above, what percent of students with good grades are heavy
users of media?
c. What percent of students with good grades are moderate users?
d. What percent of light users are students with bad grades?
e. What percent of moderate users are students with bad grades?
226
2017 AP® STATISTICS FREE-RESPONSE QUESTIONS
3. A grocery store purchases melons from two distributors, J and K. Distributor J provides melons from organic
farms. The distribution of the diameters of the melons from Distributor J is approximately normal with mean
133 millimeters (mm) and standard deviation 5 mm.
(a) For a melon selected at random from Distributor J, what is the probability that the melon will have a
diameter greater than 137 mm?
Distributor K provides melons from nonorganic farms. The probability is 0.8413 that a melon selected at random
from Distributor K will have a diameter greater than 137 mm. For all the melons at the grocery store, 70 percent
of the melons are provided by Distributor J and 30 percent are provided by Distributor K.
(b) For a melon selected at random from the grocery store, what is the probability that the melon will have a
diameter greater than 137 mm?
(c) Given that a melon selected at random from the grocery store has a diameter greater than 137 mm, what is
the probability that the melon will be from Distributor J?
227
®
2014 AP STATISTICS FREE-RESPONSE QUESTIONS
2. Nine sales representatives, 6 men and 3 women, at a small company wanted to attend a national convention.
There were only enough travel funds to send 3 people. The manager selected 3 people to attend and stated that
the people were selected at random. The 3 people selected were women. There were concerns that no men were
selected to attend the convention.
(a) Calculate the probability that randomly selecting 3 people from a group of 6 men and 3 women will result
in selecting 3 women.
(b) Based on your answer to part (a), is there reason to doubt the manager’s claim that the 3 people were
selected at random? Explain.
(c) An alternative to calculating the exact probability is to conduct a simulation to estimate the probability.
A proposed simulation process is described below.
Each trial in the simulation consists of rolling three fair, six-sided dice, one die for each
of the convention attendees. For each die, rolling a 1, 2, 3, or 4 represents selecting a
man; rolling a 5 or 6 represents selecting a woman. After 1,000 trials, the number of
times the dice indicate selecting 3 women is recorded.
Does the proposed process correctly simulate the random selection of 3 women from a group of 9 people
consisting of 6 men and 3 women? Explain why or why not.
228
Unit 4
Chapter 6
Random Variables
and Probability
Distributions
229
AP Statistics Handout: Lesson 6.1
Topics: discrete random variables, expected frequency, expected value
Lesson 6.1 Guided Notes
Discrete Random Variables
Roulette: Let X = winnings from a $100 bet on black
X=
+$100
-$100
18 Black Spaces
18 Red Spaces
1 Green Space
P(X)
In your own words, describe the difference between a regular variable and a random variable:
Expected Frequency
If I were to play roulette 1000 times, how many times would black be expected to win? Show your work.
Expected Value
If I were to play roulette 1000 times, what would my average winnings be per play? Show your work.
230
Expected Value: the _____________ value of a chance process, after repeating ___________________.
Use the space below to interpret the expected value of playing roulette (-$2.80):
Note: Losing $2.80 is _______ a possible outcome for ____________________ (+/- $100 are the only
outcomes). The expected values tells you the average over many trials, not the result of a single trial.
Calculating expected values: ________________ the outcomes (multiply) by their probabilities, then
add together to find the expected value.
Example 1: You play a coin toss game. Heads means you get $100, tails means you lose $100. Find the
expected value.
Example 2: You play a coin toss game. Heads means you get $300, tails means you lose $100. Find the
expected value.
Higher positive ___________ leads
to higher expected value!
Example 3: You play a coin toss game. The coin is weighted so that there is a 75% chance of tails. Heads
means you get $100, tails means you lose $100. Find the expected value.
Higher ___________________ of
negative value leads to a lower
expected value!
Example 4: You play a coin toss game. The coin is weighted so that there is a 75% chance of tails. Heads
means you get $300, tails means you lose $100. Find the expected value.
Expected values are weighted
averages: in this case, it
________________________.
231
Expected Value - Roulette
X=
+$100
-$100
P(X)
Expected loss for 1,000 plays:
Expected casino profit for 1,000 plays:
Individuals don’t have many trials, but casinos do. They profit through ___________________ expected
values.
Lesson 6.1 Discussion
Imagine the casinos didn’t take AP Stats. They come up with this game for high-rollers: You roll a 6-sided
dice. If you roll 5-6, you get $1,000,000. If you roll 1-4, you have to pay $300,000.
X=
+$1,000,000
-$300,000
Discussion Question: If you could play this game only
once, would you play? Why or why not? Mention the
expected value in your answer.
P(X)
232
How many children are in your family?
Count up the number of children in your family (including yourself). Be sure to include all
your stepbrothers/stepsisters and half-brothers/half-sisters.
Let X = the number of children. Suppose we choose someone from the class at random.
X
1
2
3
4
5
6
7
8
Probability
1. Is this a valid probability model? Explain.
2. Is 5.7167 a possible value for X? Explain.
3. Make a histogram to display information with X on the horizontal axis, and
describe its shape.
4. Describe in words what P ( X ³ 3) and then find P ( X ³ 3).
5. Describe in words what P( X > 3) and then find P( X > 3) .
6. Find the average of the X values.
7. Does this value tell us the average number of children in the families of students
in this class? If yes, explain. If no, why not?
233
Discrete Random Variables
Important ideas:
Check Your Understanding
Home Alone is a series of American Christmas family comedy films created by John Hughes.
There are 5 Home Alone movies. Below is a probability distribution of the number of Home Alone
movies watched by a very large sample of high school students.
Number of Home Alone
Movies Watched
Probability
0
1
2
3
4
5
0.15
0.42
0.32
?
0.02
0.01
a. Write the event “the student has seen 3 Home Alone movies” using probability notation.
Then find this probability.
b. Explain in words what P(X ≥ 3) means. What is this probability?
c. Make a histogram of the probability distribution. Describe its shape.
d. Calculate and interpret the expected value of X.
234
How much do you get paid?
Suppose you got a new job and each day your boss (Mrs. Gallas) draws a slip of paper
from a bag to determine your wage for the day. Let the random variable X = daily wage ($
per hour).
1. What is your wage for the day?
and complete the table below.
X
1
5
Add your data to the table on the board
7
10
15
25
Probability
2. Calculate and interpret the expected value of X.
Value
3. Recall from chapter 1 that standard
deviation tells us the typical distance from
the mean. Complete the table to calculate
the standard deviation for the probability
distribution.
Distance
from mean
(Distance
from mean)2
1
5
7
4. Interpret the standard deviation.
10
15
25
5. Mrs. Gallas decides she would rather
assign wages so that employees could get
any amount from $10 to $20 and all are
equally likely. Draw a graph to represent
this probability distribution.
Total =
SD =
6. What is the probability that an employee makes between $12 and $12.50?
235
Weighted
(Distance
from the
mean)2
Probability and Continuous Random Variables
Important ideas:
Check Your Understanding
Among those who play Minecraft, the amount of time they spend playing per day is
approximately Normally distributed with mean μ = 150 minutes and standard deviation σ = 42.7
minutes. Suppose we choose a Minecraft player at random and let Y = the amount of time they
spend playing Minecraft (in minutes).
a. What type of variable is Y, discrete or continuous? Explain.
b. Interpret the standard deviation.
c. Find "($ ≤ 90). Interpret this value.
d. Find "(45 ≤ $ ≤ 90). Interpret this value.
236
6.1 PRACTICE!
NHL Goals
In 2010, there were 1319 games played in the National Hockey League’s regular season. Imagine selecting one of these
games at random and then randomly selecting one of the two teams that played in the game. Define the random
variable X = number of goals scored by a randomly selected team in a randomly selected game. The table below gives
the probability distribution of X:
Goals:
0
1
2
3
4
5
6
7
8
9
Probability: 0.061 0.154 0.228 0.229 0.173 0.094 0.041 0.015 0.004 0.001
(a) Show that the probability distribution for X is legitimate.
(b) Make a histogram of the probability distribution. Describe what you see.
(c) What is the probability that the number of goals scored by a randomly selected team in a randomly selected game is
at least 6?
237
NHL Goals, continued.
Previously, we defined the random variable X to be the number of goals scored by a randomly selected team in a
randomly selected game. The table below gives the probability distribution of X:
Goals:
0
1
2
3
4
5
6
7
8
9
Probability: 0.061 0.154 0.228 0.229 0.173 0.094 0.041 0.015 0.004 0.001
(d): Compute the mean of the random variable X and interpret this value in context.
(e): Compute and interpret the standard deviation of the random variable X.
Weights of Three-Year-Old Females
2. The weights of three-year-old females closely follow a Normal distribution with a mean of  = 30.7 pounds and a
standard deviation of 3.6 pounds. Randomly choose one three-year-old female and call her weight X.
A. Find the probability that the randomly selected three-year-old female weighs at least 30 pounds.
B. Should the pediatrician be concerned if a 3 year old girl only weighs 25 pounds?
238
AP Statistics Handout: Lesson 6.2
Topics: variance & standard deviation, transforming & combining random variables
Lesson 6.2 Guided Notes
Variance and Standard Deviation of Random Variables
Standard Deviation vs. Variance
Standard Deviation:
______________________
from mean
∑(𝑥𝑖 − 𝑥̅ )2
20922.92
𝝈=√
=√
= √1902.08 = 43.61
𝑛−1
11
Variance:
______________________________
measure of spread
∑(𝑥𝑖 − 𝑥̅ )2 20922.92
𝝈𝟐 =
=
= 1902.08
𝑛−1
11
Mathematical Relationship
𝝈𝟐 = 𝑽𝒂𝒓
𝝈 = √𝑽𝒂𝒓
Variance for random variables
Roulette: Let X = winnings from a $100 bet on black
X=
+$100
-$100
P(X)
E(X) = 𝜇𝑋 = −2.80
Find the variance and the standard deviation of X:
Var(X) = ∑(𝑋𝑖 − 𝜇𝑋 )2 𝑝𝑖
239
18 Black Spaces
18 Red Spaces
1 Green Space
“A Good Game”
Imagine the casinos didn’t take AP Stats. They come up with this game for high-rollers: You roll a 6-sided
dice. If you roll 5-6, you get $1,000,000. If you roll 1-4, you have to pay $300,000.
X=
+$1,000,000
If you could play this game only once, would you play?
Why or why not?
-$300,000
P(X)
𝜇𝑋 ≈ $132,900
𝜎𝑋 ≈ $612,826
Transforming Random Variables
Transforming Roulette: Let X - 5 = winnings from a $100 bet on black, with $5 fee to play.
X-5=
P(X)
18
37
= .486
19
37
= .514
18 Black Spaces
18 Red Spaces
1 Green Space
a) Find the expected value for playing this game.
E(X ± constant) = _____________________
b) Find the variance and standard deviation for this game.
Adding/subtracting all outcomes by a constant ______________________________ the variation
between outcomes!
When adding or subtracting random variables by a constant…
1. Center: adds/subtracts by that constant amount.
2. Spread: remains the ___________.
3. Shape: remains the ___________.
240
Multiplying a Random Variable by a Constant
E(2X) = 2𝜇𝑋
Standard Deivation(2X) = 2𝜎𝑋
Var(2X) =
When multiplying or dividing random variables by a constant…
1. µ: multiplies/divides by that constant.
2. 𝝈: multiplies/divides by that constant.
𝝈𝟐 : multiplies/divides by that constant ___________________.
3. Shape: remains the ____________.
Combining Random Variables
Adding Random Variables
Usually, people who play in the casinos play more than just roulette. Say that you play the slots and you
put in $20. The slots are programmed so that you have a 20% chance of doubling your money and an
80% chance of losing your money. Here’s is the probability model for this random variable:
Y=
+$20
-$20
P(Y)
0.20
0.80
Y = winnings from slot machine.
Find the expected value and standard deviation of playing both roulette ($100 bet) and the slot machine
($20 bet). Let Z = X+ Y (winnings/losses from both games).
What if we just added the standard deviations instead of the variances?
• You __________________ standard deviations! You must use variances and square root at end.
241
Subtracting Random Variables
Let X be a random variable with 𝜇𝑋 = 2 and 𝜎𝑋 2 = 4. Let Y be a random variable with 𝜇𝑌 = 3 and 𝜎𝑌 2 = 25
Let Z = X-Y.
a) What is 𝛍𝐙 ?
When subtracting random variables, ___________________________!
b) What is 𝝈𝒁 ?
When subtracting random variables, ____________________________________!
Lesson 6.2 Discussion
You are at the casino and decide to ignore your stats teacher and play high-stakes roulette. You have
one of two options:
1. Place a $200,000 bet on red for one turn
2. Place a $100,000 bet on red for each of two separate turns
Discussion Question: Which strategy would you take? Does it matter? Justify your reasoning.
242
Is it time for a raise?
Mrs. Gallas’ employees have been working very hard and it’s time she gives them a raise.
She is trying to decide if she should give everyone a $10 raise (add $10 per hour) or
double everyone’s wage (multiply by 2).
1. Copy the data collected from yesterday’s lesson below.
X
1
5
7
10
15
25
Probability
Mean:
Standard Deviation:
2. To make a decision about what raise should be given, complete the tables below and
calculate the new mean and standard deviation using an applet or your calculator.
a. Option 1: Add $10 per hour to all employees
Xà Old Wage
1
5
7
10
15
25
Y à New Wage
Probability
Mean:
Standard Deviation:
How did adding a constant affect the mean and standard deviation?
b. Option 2: Double the original wage of all employees
X à Old Wage
1
5
7
10
15
Z à New Wage
Probability
Mean:
Standard Deviation:
How did multiplying by a constant affect the mean and standard deviation?
3. Which option would you prefer? Why?
243
25
Transforming Probability Distributions
Important ideas:
Check Your Understanding
Let X = the number billionaires in a randomly selected state. Based on current records, the
probability distribution of X is as follows:
Number of
0
1
2
3
4
5
6
8
9
10 11 12
13 17 56 58 118 165
Billionaires
Probability 0.10 0.24 0.06 0.06 0.04 0.08 0.04 0.04 0.06 0.08 0.02 0.02 0.02 0.06 0.02 0.02 0.02 0.02
The random variable X has mean 𝜇! = 12.68 and standard deviation 𝜎! = 29.02. Suppose a law
is passed requiring each billionaire to pay $1,000,000 to their state. Let Y = the money received
by a randomly selected state.
1. Consider the graph of the probability distribution of X and a separate graph of the
probability distribution of Y. How would their shapes compare?
2. Find the mean of Y.
3. Calculate and interpret the standard deviation of Y.
4. Each state agrees to invest $500,000 to improve roads. Therefore, the net amount (N) of
money a randomly selected state has after the billionaire tax is N = Y – $500,000.
Describe the shape, mean, and standard deviation of the probability distribution of N.
244
How much will you make next year?
After much thought Mrs. Gallas has finally decided on permanent employee wages which
are randomly assigned using the probability distribution X given below. Additionally, at the
end of every year she gives her employees an hourly raise. The bonuses are assigned
randomly according to the probability distribution Y given below. Assume X and Y are
independent.
1. Find the mean, variance and standard deviation of the probability distribution of X, the
hourly wages.
9
12
15
X
Probability
Mean:
0.30
0.45
Variance:
0.25
Standard Deviation:
2. Find the mean, variance and standard deviation of the probability distribution of Y, the
annual hourly raise.
Y
Probability
Mean:
$1
$3
0.70
0.30
Variance:
Standard Deviation:
3. Let N = the new hourly wage for the upcoming year (X + Y).
a. What are all the possible new hourly wages for the new year?
b. What is the probability of an employee being assigned a $9 wage AND a $1 raise?
Show your work.
c. Complete the table below for the probability distribution of N = X + Y and find the mean
and standard deviation.
N
Probability
Mean:
Variance:
Standard Deviation:
d. If N = X + Y, complete the following in terms of X and Y:
𝜇! =
𝜎! =
245
Combining Probability Distributions
Important ideas:
Check Your Understanding
Mrs. Chauvet recently had twins. Let X = the number of diaper changes per day for Alyse and Y =
the number of diaper changes per day for Jocelyn. Based on a few weeks of careful records, the
probability distributions of X and Y are as follows:
Number of diapers
changed xi
Probability pi
Mean: µ X = 4.75
3
4
5
0.05 0.25 0.60
SD: s X = 0.698
6
0.10
Number of diapers
changed yi
Probability pi
Mean: µY = 4.9
3
4
5
6
0.05 0.20 0.55 0.20
SD: s Y = 0.768
Define T = X + Y. Assume that X and Y are independent.
a. Find and interpret 𝜇 " .
b. Calculate and interpret 𝜎" .
c. Alyse wears Diaper size 1, which cost $0.238 per diaper and Jocelyn wears Diaper size 2
which cost $0.2975 per diaper. Find the mean and standard deviation of Mrs. Chauvet’s total
diaper cost per day.
246
6.2 PRACTICE!
Study Habits
The academic motivation and study habits of female students as a group are better than those of males. The Survey of
Study Habits and Attitudes (SSHA) is a psychological test that measures these factors. The distribution of SSHA scores
among the women at a college has mean 120 and standard deviation 28, and the distribution of scores among male
students has mean 105 and standard deviation 35. You select a single male student and a single female student at
random and give them the SSHA test.
(a) Explain why it is reasonable to assume that the scores of the two students are independent.
(b) What are the expected value and standard deviation of the difference (female minus male) between their scores?
(c) From the information given, can you find the probability that the woman chosen scores higher than the man? If so,
find this probability. If not, explain why you cannot.
247
Swim Team
Alonzo & Tracy Mourning Sr. High School has one of the best women’s swimming team in the region. The 400-meter
freestyle relay team is undefeated this year. In the 400-meter freestyle relay, each swimmer swims 100 meters. The
times, in seconds, for the four swimmers this season are approximately Normally distributed with means and standard
deviations as shown. Assuming that the swimmer’s individual times are independent, find the probability that the total
team time in the 400-meter freestyle relay is less than 220 seconds. Show your work!
248
AP Statistics Handout: Lesson 5.7
Topics: geometric and binomial random variables
Lesson 5.7 Guided Notes
Free Throw Accuracy
49.7%
Free Throw Accuracy
61.3%
Wilt Chamberlain: traditional
(all seasons except ’61-’62)
Wilt Chamberlain: granny-style
(1961-1962 season)
Today’s Analysis: Why did players stop shooting “granny-style” free throws?
The Geometric Distribution
Story of Geometric:
• ________________ trials (success/fail → make/miss)
• Looking for ______________________
• Each trial is _____________________
When Wilt Chamberlain shot “granny-style,” he made 61.3% of his free-throws. Say that Wilt shoots
granny-style until he makes one free-throw. Assuming each shot is independent, what is the probability
that his first make comes on his 5th shot? Show calculations here:
249
Geometric PDF formula: What is the probability that first success in on xth attempt? (let p = probability
of success)
𝑋~𝐺𝑒𝑜𝑚(𝑝)
𝑃(𝑋 = 𝑥) = (1 − 𝑝)
Called the “probability density function” or _______
𝑥−1
(𝑝)
PDF vs. CDF
• PDF situation: What is the probability that Wilt’s first made shot is on his 3rd attempt?
“Probability” means
_____________ probability
•
Cumulative Distribution Function (CDF) situation: What is the probability that his first made
shot is on his 3rd attempt or earlier?
“Cumulative” means
_____________________
Using the calculator: Find the probability that his first made shot is on the 100th attempt or earlier.
2nd → DISTR → E: GeometPDF
2nd → DISTR → F: GeometCDF
The Binomial Distribution
When Wilt Chamberlain shot “granny-style,” he made 61.3% of his free-throws. Say that Wilt shoots 8
consecutive granny-style free-throws. Assuming each shot is independent, what is the probability that
he makes exactly 3 of them?
Story of Binomial:
• __________________ trials (success/fail → make/miss)
• Each trial is ____________________________
• ______________ number of trials (8)
• ____________ # of successes (3)
250
Binomial PDF formula: What is the probability of making x out of n shots? (let p = probability of success)
𝑋~𝐵𝑖𝑛𝑜𝑚(𝑛, 𝑝)
𝑛
𝑃(𝑋 = 𝑥) = ( ) 𝑝 𝑥 (1 − 𝑝)𝑛−𝑥
𝑥
a) What is the probability of making 3 out of 8 shots?
b) What is the probability of making at most 3 out of 8 shots?
2nd → DISTR → A: BinomPDF
2nd → DISTR → B: BinomCDF
c) What is the probability of making at least 4 out of 8 shots?
Lesson 5.7 Discussion
Discussion Question: Why do you believe shorter players tend to be better at free throws? Give a
statistical (not physics-based) reason for your answer.
251
Is it smart to foul at the end of the game?
In the 2005 Conference USA basketball tournament, Memphis trailed Louisville by two
points. At the buzzer, Memphis’s Darius Washington attempted a 3-pointer; he missed but
was fouled, and went to the line for three free throws. Each made free throw is worth 1
point. Was it smart to foul?
1. What are all the possible ways the shots could fall (e.g. make-miss-miss, etc.)?
2. Darius Washington was a 72% free-throw shooter. Find the probability that
Memphis will win, lose or go to overtime. When you have found the probabilities
put them in the table in #3.
3. Prior to watching each shot, calculate the probability that Memphis wins the
game in regulation, loses the game in regulation, or sends the game into
overtime.
4. Washington is a 40% 3-point shooter. Do you think Louisville was smart to foul?
Why or why not?
252
Binomial Random Variables
Important ideas:
Check Your Understanding
1. For each of the following situations, determine whether or not the given random variable has
a binomial distribution. Justify your answer.
a. You play a game of Whack-a-Mole. From playing this game in the past, you know that
you have an 80% probability of whacking the mole before it drops back into its hole. The
moles pop up randomly and your ability to whack any particular mole is not affected by
whether or not you whacked the previous mole. There are 20 moles to be whacked
during one round of the game. Let X = the number of moles you are able to whack.
b. Next you play Skee Ball. You know that you have a 10% probability of getting any
given ball in the 10,000-point hole. Let Y = the number of balls you must roll until you
get one in the 10,000-point hole.
2. Now you play a game called Tsunami Duck Pond. In Tsunami Duck Pond there are 100
ducks that get pummeled by tidal waves. You have to reach your hand into the tsunami
and select a duck. If there is a star on the bottom of the duck, you win. The game
claims to have 20 ducks with stars among the 100 ducks. After each round you must
place the duck back in the tumultuous water. Let W = the number of times you win if you
play this game 10 times.
a. Explain why W is a binomial random variable.
b. Find the probability that you win exactly 3 times.
253
Lesson 6.3: Day 2: Will the EKHS girls’ soccer team win?
When the time runs out in a soccer game and the score is tied, the game will go to a
shootout. Each team gets to choose 5 players to kick penalty kicks. Whichever team
makes the most penalty kicks wins. If the SAHS girls’ soccer team makes 60% of
their penalty kicks, what are the chances they will win the game?
1. Is this a binomial setting? Explain.
2. Fill in the table below showing the probability of making X penalty kicks.
Goals (X)
0
1
2
3
4
5
Probability
3. Find and interpret the mean of the probability distribution. Show your work.
4. Find and interpret the standard deviation of the distribution.
5. What is the probability that the team scores at least one goal?
6. If the other team is expected to make 3 goals, what is the probability that the SAHS
girls’ team wins?
254
Lesson 6.3 Day 2– Describing Binomial Distributions
Important ideas:
Check Your Understanding
Mrs. Mason’s class is very difficult. It’s so hard that when she gave a pop quiz recently, the
students just guessed on every question! Each student in the class guesses an answer from A
through E on each of the 10 multiple-choice questions. Emily is one of the students in this
class. Let Y = the number of questions that Emily answers correctly.
1. Does this setting represent a binomial distribution? Explain.
2. Use technology to make a probability distribution of Y. Describe its shape if it was a histogram.
3. Calculate and interpret the mean of Y.
4. Calculate and interpret the standard deviation of Y.
255
Where are all the Orange Skittles?
Mrs. Mason loves orange Skittles, but the last time she got a fun size bag there were no
orange Skittles! The bag was manufactured at the Skittles factory where 10,000 Skittles
are made each hour. 20% of the Skittles are orange. To make a fun size bag, 10 Skittles
are chosen for each bag. What are the chances of getting a fun size bag with no orange
Skittles?
1. Is this a binomial setting? Explain.
2. Find the probability of getting 0 orange Skittles in a group of 10 from the factory. Write
the binomial formula from the formula sheet.
3. Find the probability of getting at most 5 orange Skittles in a group of 10 from the factory.
To ensure that she gets more orange Skittles, Mrs. Mason buys a jumbo bag of Skittles
which contains 900 Skittles. Let X = number of orange Skittles in a bag of 900 Skittles. Use
a binomial distribution to model the situation.
4. Find the mean and standard deviation of X.
5. What is the probability of getting at most 150 orange Skittles?
6. If we were to make a histogram of X, what do you think the shape would be?
256
Binomial Distributions
Big ideas:
Check Your Understanding
In a survey of 500 U.S. teenagers aged 14 to 18, subjects were asked a variety of questions
about personal finance. One question asked whether teens had a debit card. Suppose that
exactly 12% of teens aged 14 to 18 have debit cards. Let X = the number of teens in a random
sample of size 500 who have a debit card.
1. Is this binomial? Estimate the probability that 50 or fewer teens in the sample have debit cards.
2. Find and interpret in context: P(x>100).
3. Find and interpret in context: p(60<x<100):
3. Find and interpret the mean and standard deviation::
257
6.3 PRACTICE!
1. Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with
possible values of 1 to 5. Assume that X is a random variable representing the pain score for a randomly elected patient.
The following table gives part of the probability distribution for X.
(a) Find P(X = 5).
(b) Find the probability that the pain score is less than 3.
(c) Find the probability that the pain score is greater than 3.
(d) Find the mean μ for this distribution.
2. Amarillo Slim, a professional dart player, has an 80% chance of hitting the bull’s-eye on a dartboard with any throw.
Suppose that he throws 10 darts, one at a time, at the dartboard.
(a) Find the probability that Slim hits the bull’s-eye exactly six times.
(b) Find the probability that he hits the bull’s-eye at least four times.
(c) Compute the expected number of bull’s-eyes in 10 throws.
We will add more to this problem set LATER :)
258
(d) Find the probability that Slim’s first bull’s-eye occurs on the fourth throw.
(e) Find the probability that it takes Amarillo more than 2 throws to hit the bullseye.
3. Harlan comes to class one day, totally unprepared for a pop quiz consisting of ten multiple-choice questions. Each
question has five answer choices, and Harlan answers each question randomly.
(a) Find the probability that Harlan’s gets more than 5 questions right out of 10.
(b) Find the probability that Harlan’s first correct answer occurs after the fourth question.
(c) Find the expected number of questions required for Harlan to get his first correct answer.
(d) Find the probability that Harlan guesses more answers correctly than would be expected by chance.
259
6.3 PRACTICE! #2
For each problem, be sure that the situation fits the criteria for binomial distributions. If so, answer the questions (show
the formula) and then find the mean and standard deviation of the distribution.
1) 80% of the graduates of Northeast High who apply to Penn State University are admitted. Last year, there were 6
graduates from Northeast who applied to Penn State. What is the probability that
a) 4 were admitted
b) more than 4 were admitted
Mean of distribution: ____________
Standard deviation of distribution: ____________
2) Tires from the Apex Tire Corp. are traditionally 5% defective. A truck carries 10 tires, 8 in use and 2 spares. If 10 tires
are chosen from Apex, what is the probability that not more than two defective tires are chosen.
Mean of distribution: ____________
Standard deviation of distribution: ____________
3) Studies indicate that in 70% of the families of Blue Bell, both the husband and wife work. If 7 families are randomly
selected from Blue Bell, what is the probability that
a) exactly 4 of them work.
b) more than 4 work
Mean of distribution: ____________
Standard deviation of distribution: ____________
260
4) According to the National Institute of Health, 32% of all women will suffer at least one hip fracture because of
osteoporosis by the age of 90. If 10 women aged 90 are selected at random, find the probability that
a) 2 or more of them suffer/will suffer at least one hip fracture
b) none of them suffer/will suffer a hip fracture
Mean of distribution: ____________
Standard deviation of distribution: ____________
5) According to FBI statistics, only 52% of all rape cases are reported to the police. If 10 rape cases are randomly
selected, what is the probability that at least one is reported to the police?
Mean of distribution: ____________
6) In a school, typically only
1
of
10
Standard deviation of distribution: ____________
the student body returns surveys. 20 students are chosen randomly to receive a
survey. What is the probability that
a) they get no surveys back.
b) they get more 4 or more surveys back.
Mean of distribution: ____________
Standard deviation of distribution: ____________
7) The probability that a driver making a gas purchase will pay by credit card is 0.60. If 50 cars pull up to the station to
buy gas, what is the probability that at least half of the drivers will pay by credit card?
Mean of distribution: ____________
Standard deviation of distribution: ____________
261
262
Strategy:
Example problem where this
strategy would be useful
PROBABILITY STRATEGIES
What this strategy looks like:
Mutually Exclusive versus Independent
Definition: MUTUALLY EXCLUSIVE EVENTS (Disjoint):
Definition: INDEPENDENT EVENTS:
Create an example of two events A and B where:
1. Events A and B are MUTUALLY EXCLUSIVE (Disjoint)
2. Events A and B are INDEPENDENT.
3. Events A and B are MUTUALLY EXCLUSIVE and INDEPENDENT.
4. Events A and B are MUTUALLY EXCLUSIVE and DEPENDENT.
5. Events A and B are NOT MUTUALLY EXCLUSIVE and INDEPENDENT.
6. Events A and B are NOT MUTUALLY EXCLUSIVE and DEPENDENT.
263
How Many Bottle Flips to Go Viral?
Scene: The senior talent show is scheduled to take place this afternoon. Michael Senatore is in
stats class practicing his act - bottle flipping. He tells his friends that the probability of successfully
landing the water bottle right side up is 20%. Assume that each water bottle flip is independent.
1. If he flips the bottle ten times, how many do you expect to land right side up? Justify.
2. What is the probability that he makes exactly two flips?
3. What is the probability that he makes less than two flips?
4. What is the probability that he makes at most two flips?
After the first ten flips, Michael wants to do ten more, but his teacher is getting annoyed with all the noise
and tells him he must stop flipping once the bottle lands right side up.
5. Find the probability that he lands it on the first try.
6. Find the probability that he fails on the first try and lands it on the second try.
7. Find the probability that he fails the first two and lands it on the third try.
8. Find the probability that he fails the first three and lands it on the fourth try.
Let X = the number of flips it takes for Michael to land the bottle right side it up. Fill in the table below.
X
1
2
3
4
5
6
P(X)
9. Find the probability that he lands it on the tenth try. Write a generic rule for finding the probability
that he first lands the bottle on the kth flip.
10. How many flips do you expect it to take for the bottle to land right side up? Why?
264
The Geometric Distribution
Important ideas:
Check Your Understanding
Mason never has a pencil when test day rolls around. Because the classmates are tired of having to
supply pencils for Mason, only 15% of students will give Mason a pencil when asked. Today is test
day and Mason begins asking randomly selected students for a pencil. Let Y = the number of
students Mason asks until he finds someone who will give him a pencil.
a. Describe this probability distribution. Be sure to check the appropriate conditions.
b. What is the probability that the third person asked is the first person who gives him a pencil?
c. What is the probability that Mason gets a pencil by the third person he would ask?
d. How many people should Mason expect to ask before getting a pencil?
e. Should Mason be surprised if Mason didn’t receive a pencil until he asked at least 10 people?
Calculate a probability to justify your answer.
f. Find the standard deviation for the distribution of the number of people he asks for a pencil before
finding someone who will give him one.
Thank you to AP Stats teachers Kelly Pendleton and Alana Braland for this lesson
265
3. A medical researcher surveyed a large group of men and women about whether they take medicine as prescribed.
The responses were categorized as never, sometimes, or always. The relative frequency of each category is
shown in the table.
Men
Women
Total
Never
0.0564
0.0636
0.1200
Sometimes
0.2016
0.1384
0.3400
Always
0.2120
0.3280
0.5400
Total
0.4700
0.5300
1.0000
(a) One person from those surveyed will be selected at random.
(i) What is the probability that the person selected will be someone whose response is never and who is a
woman?
(ii) What is the probability that the person selected will be someone whose response is never or who is a
woman?
(iii) What is the probability that the person selected will be someone whose response is never given that the
person is a woman?
(b) For the people surveyed, are the events of being a person whose response is never and being a woman
independent? Justify your answer.
(c) Assume that, in a large population, the probability that a person will always take medicine as prescribed
is 0.54. If 5 people are selected at random from the population, what is the probability that at least 4 of
the people selected will always take medicine as prescribed? Support your answer.
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5. A company that manufactures smartphones developed a new battery that has a longer life span than that of a
traditional battery. From the date of purchase of a smartphone, the distribution of the life span of the new battery
is approximately normal with mean 30 months and standard deviation 8 months. For the price of $50, the
company offers a two-year warranty on the new battery for customers who purchase a smartphone. The warranty
guarantees that the smartphone will be replaced at no cost to the customer if the battery no longer works within
24 months from the date of purchase.
(a) In how many months from the date of purchase is it expected that 25 percent of the batteries will no longer
work? Justify your answer.
(b) Suppose one customer who purchases the warranty is selected at random. What is the probability that the
customer selected will require a replacement within 24 months from the date of purchase because the battery
no longer works?
(c) The company has a gain of $50 for each customer who purchases a warranty but does not require
a replacement. The company has a loss (negative gain) of $150 for each customer who purchases a warranty
and does require a replacement. What is the expected value of the gain for the company for each warranty
purchased?
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268
Unit 5
Chapter 7
Collecting Data
269
AP Statistics Handout: Lesson 7.1
Topics: parameters vs. statistics, sampling distributions, accuracy vs. precision
Lesson 7.1 Guided Notes
Parameters vs. Statistics
Inference: ________________________ a characteristic of a population using sample statistics.
Population
Sample
Sample: a ____________________ of a larger population.
A ______________________ is a number that describes some characteristic of the population.
A ____________________ is a number that describes some characteristic of a sample.
Example: The Business Owner Problem
A business owner wants to know the average household income in her business’s zip code. She
randomly samples 80 households in the zip code and finds their mean income to be $46,144.
Population:
Parameter:
Sample:
Statistic:
Sampling Distributions
Sampling distribution: the ________________________________ from all possible samples (of the same
size) from a population.
•
Sampling distributions are made of statistics (means, proportions, Q3’s, etc.), ___________
_____________________.
270
____________________
Data
(Data – incomes)
Data
____________________
(Data – incomes)
Means
ഥ
𝒙
ഥ𝒙
ഥ
𝒙
ഥ𝒙
ഥ
𝒙
ഥ𝒙
ഥ𝒙
ഥ𝒙
ഥ
𝒙
ഥ
𝒙ഥ
𝒙ഥ
𝒙ഥ
𝒙
ഥ𝒙
ഥ𝒙
ഥ𝒙
ഥ𝒙
ഥ𝒙
ഥ
𝒙
____________________
(Statistics – means of incomes)
Accuracy (bias) vs. Precision (variation)
How do you avoid bias?
(how do you make it accurate?)
Sample __________________
How do you reduce variability? (how do
you make it more precise?)
_____________________ sample size (n)
Sample size: n = 5
Sample size: n = 25
_______________
(centered at true mean)
____________
(more precise)
Extension question: Why do higher
sample sizes lead to more precise
estimates?
271
Lesson 7.1 Discussion
The German Tank Problem
Let’s say German tanks with the following July serial numbers were captured
during combat. Assume it’s a random sample of the tanks produced:
038, 072, 121, 158, 206
a) Identify the following:
Population:
Parameter:
Sample:
Statistic:
b) Discussion Question: Calculate a statistic, using this sample, to estimate the total number of tanks
produced. Report your result and justify your process.
272
What was the average for the Chapter 6 test?
How did the Chapter 6 test go? Today, we will be taking a sample from a population. We
will use the average from the sample to estimate the average for the population.
Let’s start with a very simple example. My 5th hour is very small. There were only 4 students
who took the chapter 6 test. Their scores were: 60 70 80 90.
1. Make a dotplot of the population distribution.
2. Take a sample of any 2 of the scores. Find the mean of your sample.
3. Figure out all of the possible samples of size 2. Calculate a sample mean for each
sample of 2.
4. Make a dotplot using each of the means you found in #3.
5. What is the mean of the population? Label this on the dotplot above.
273
What is a Sampling Distribution? Day 1
Important ideas:
Check Your Understanding
The James family has five children: Jocelyn (age 8), Alyse (age 8), Michael (age 14), Erica (age
16), and Sarah (age 18).
a. Complete the table by listing the 10 possible samples of size n = 2 from this population and
calculate the sample mean age for each sample. The first column is completed for you.
Sample
J, A
n=2
8
x
b. Create a sampling distribution of the sample mean age of samples of size 2.
c. What is the mean of the sampling distribution of the sample mean? What is the mean of the
population?
d. Is the sample mean an unbiased estimator of the population mean? Justify your answer.
e. Suppose we had taken samples of size 3 instead of size 2. Would the variability of the sampling
distribution of the sample mean be larger, smaller, or about the same? Justify your answer.
274
What was the real average for the Chapter 6 test?
How did the Chapter 6 test go? Today, we will be taking a sample from a population. We
will use the average from the sample to estimate the average for the population.
Yesterday we looked at a very small class of students as the population. In reality there were
many students who took the test. Take a random sample of 5 students and record their
scores. Then find the mean. Repeat this for a total of 4 times.
Scores:______________ Mean:______
Scores:______________ Mean:______
Scores:______________ Mean:______
Scores:______________ Mean:______
1. Write each mean on a different sticker and put the stickers in the appropriate location
on the poster at the front of the room. Copy down the dotplot that is created on the
poster.
2. What does each dot on the poster represent?
3. What do you think the true Chapter 6 test average is?
4. A sampling distribution shows the means calculated from all of the possible samples
of size 5 from the population. Is the above dotplot a sampling distribution? Explain.
5. We took a random sample of 5 test scores at Rockford high school and got a mean of
68. Is this convincing evidence that Rockford students did worse than students at our
school or is it possible the Rockford has the same average?
275
What is a Sampling Distribution? Day 2
Important ideas:
Check Your Understanding
Pennies made prior to 1982 were made of 95% copper. Because of their copper content, these pennies are
worth about $0.023 each. Pennies made after 1982 are only 2.5% copper. Jenna reads online that 13.2% of
pennies in circulation are pre-1982 copper pennies. Jenna has a large container of pennies at home. She
selects a random sample of 50 pennies from the container and finds that 11 are pre-1982 copper pennies.
Does this provide convincing evidence that the proportion of pennies in her container that are pre-1982
copper pennies is greater than 0.132?
1. Identify the population, parameter, sample and statistic.
Population:
Sample:
Parameter:
Statistic:
2. Does Jenna have some evidence that more than 13.2% of her pennies are pre-1982 copper pennies?
3. Provide two explanations for the evidence described in #2.
We used technology to simulate selecting 100 SRSs of size
n = 50 from a population of pennies in which 13.2% are pre1982 copper pennies. The dotplot shows 𝑝̂ = the sample
proportion of copper pennies for each of the 100 samples.
4. There is one dot on the graph at 0.22 (or 22%). Explain
what this dot represents.
5. Assuming that 13.2% of pennies in circulation are pre-1982 copper pennies, is it surprising to randomly
select 50 pennies for which 𝑝̂ = 11/50 = 22% or greater? Justify your answer.
6. Based on your previous answers, is there convincing evidence that more than 13.2% of pennies in
Jenna’s container are pre-1982 copper pennies? Explain your reasoning.
276
7.1 PRACTICE!
Tall Girls
According to the National Center for Health Statistics, the distribution of heights for 16-year-old females is modeled well
by a Normal density curve with mean μ = 64 inches and standard deviation σ = 2.5 inches. To see if this distribution
applies at their high school, an AP Statistics class takes an SRS of 20 of the 300 16-year-old females at the school and
measures their heights. What values of the sample mean X would be consistent with the population distribution being
N(64, 2.5)? To find out, we used Fathom software to simulate choosing 250 SRSs of size n = 20 students from a
population that is N(64, 2.5). The figure below is a dotplot of the sample mean height X of the students in the sample.
(a) Is this the sampling distribution of X? Justify your answer.
(b) Describe the distribution. Are there any obvious outliers?
(c) Suppose that the average height of the 20 girls in the class’s actual sample is X = 64.7. What would you conclude
about the population mean height μ for the 16-year-old females at the school? Explain.
277
Choosing Cards
We used Fathom (statistical computer software) to simulate choosing 500 SRSs of size 5 from the deck of cards
described in the Alternate Activity on the previous page. The graph below shows the distribution of the sample
median for these 500 samples.
(a) Is this the sampling distribution of the sample median? Justify your answer.
(b) Suppose that another student prepared a different deck of cards and claimed that it was exactly the same as the one
used in the activity. However, when you took an SRS of size 5, the median was 4. Does this provide convincing evidence
that the student’s deck is different?
278
AP Statistics Handout: Lesson 7.2
Topics: p vs. 𝑝̂, sampling distribution for a proportion, spread and sample size
Lesson 7.2 Guided Notes
Why you care about proportions
What is a proportion?
Proportion: the _________ of a binary occurrence
• Written as fraction, decimal, or %
p vs. 𝑝̂
In your own words, describe why it’s difficult to obtain unbiased estimates about immigration in the
United States:
The EU Migration Debate and Gender of Immigrants
Vital International
AP Photo/Michael Svarnias
Sympathy for Refugees
Depictions of migrants in the EU
range from sympathetic portrayals
of families escaping violence to
fear-inducing portrayals of groups
of single adult men illegally
crossing borders.
Fear for Jobs & Terrorism
“As you can see from this picture, most of the people coming are young males and,
yes, they may be coming from countries that are not in a very happy state, they may
be coming from places that are poorer than us, but the EU has made a fundamental
error that risks the security of everybody.”
– Nigel Farage, 2016 (U.K.)
Speaking about EU migration: “I said to myself, ‘Wow. They’re all men.’ You look at
it. There are so few women and there are so few children. And not only are they
men, they’re young men. And they’re strong as can be—they’re tough looking
cookies. I say, what’s going on here?”
– Trump, during 2015 campaign speech
279
The EU Migration Study – Spain
Trump/Farage statistical claim: 60% of migrants are male
To test this claim, we will use data from Spain, which has the best records on EU migrants (both
documented and undocumented migrants). According to a report* in Comparative Migration Studies,
Spain has a “nation-wide rolling local population register, which provides a comprehensive register of
the population, also including unauthorized immigrants.”
• The Immigrant Citizens Survey** conducted a random sample of this database, surveying recent
immigrants to Spain who were 15 and older.
𝑝 = ___________________ proportion
• Parameter
𝑝̂ = _________________ proportion
• Statistic
• Estimator of p
The Immigrant Citizens Survey conducted a random sample of 15+ year-old recent migrants in Spain’s
comprehensive database. In Spain’s immigrant hub, the city of Madrid, 257 of the 583 immigrants
sampled were male.
Population:
Parameter:
Sample:
Statistic:
*Reichel and Morales, “Surveying immigrants without sampling frames – evaluating the success of alternative field methods,” Comparative
Migration Studies (2017) 5:1 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5209407/pdf/40878_2016_Article_44.pdf
**Huddleston, Thomas and Dag Tjaden, Jasper. ”Immigrant Citizens Survey: How immigrants experience integration in 15 European cities,” King
Baudouin Foundation and The Migration Policy Group. May 2012.
Sampling Distribution for a Proportion
Welcome to Theory Land – “The not real place.”
Population of all migrants
Realm of Perfect Parameters
Realm of Sucky Statistics (n = 30)
What proportion of all Madrid immigrants are male?
Let’s get a sample!
p = 0.60
Random sample possibility #1:
Random sample possibility #2
Random sample possibility #3:
280
Sampling Distribution for a Proportion - the formula
Under certain conditions:
𝑝̂ ~ Normal(𝜇𝑝̂ = 𝑝, 𝜎𝑝̂ = √
Centered at true proportion, meaning the
estimates are _______________.
𝑝(1−𝑝)
𝑛
)
We _____________________, so spread
affected by sample size. Higher sample
size means _______________________.
Assume the Trump/Farage statistical claim is true: 60% of all migrants to the EU are male.
a) Find the sampling distribution for 𝑝̂ (sample proportion of males) among 30 randomly selected
migrants (n = 30). Assume all conditions are met:
b) If the Trump/Farage claim was true, what is the probability
that a random sample of 30 migrants would find 15 or fewer males?
Spread and Sample Size
Again, assume the Trump/Farage statistical claim is true: 60% of all migrants to the EU are male.
However, now we’ll use the sample size from the EU Migration Study: n = 583. Find the sampling
distribution for 𝑝̂ (sample proportion of males) among 583 randomly selected migrants. Assume all
conditions are met:
Raising the sample size makes estimates
________________________.
281
What’s the proportion of orange Reese’s Pieces?
If we take a sample of Reese’s Pieces, what proportion of the candies will be orange?
Suppose a large bag of Reese’s Pieces has 1000 pieces. The manufacturer says that
exactly 40% of the candies are orange. If we select a sample of 50 pieces, how many will be
orange? Let X = the number of orange candies in the sample.
1. What type of probability distribution does X have? Justify.
2. Draw a sample of 50 Reese’s Pieces using the applet. How many pieces were
orange? Repeat this 5 times. Write the values below.
3. Write the values on sticker dots and add it to the dotplot on the board. Sketch the
dotplot below.
4. What does each dot represent?
5. What is the mean and the standard deviation for the distribution of X? Show work.
6. What is the approximate shape of the sampling distribution for X? Explain and
sketch it below.
282
Instead of finding the number of candies that are orange, we will now find the proportion of
candies that are orange.
7. Use your samples from #2 and turn each number of orange candies into the
proportion of orange candies in the sample ( p̂ ). Write the proportions below and
add them to the second dotplot on the board.
8. Sketch the dotplot below.
9. What does each dot represent?
10. Find the new mean and standard deviation. Show work.
11. What is the approximate shape of the sampling distribution for !̂ ? Explain and
sketch it below.
12. We know that bags of Reese’s Pieces contain exactly 40% that are orange. If we
select a random sample of 50 candies, what is the probability that the sample
proportion will be 50% or greater?
283
The Sampling Distribution of !̂
Important ideas:
Check Your Understanding
According to the American Dental Association, 8% of adults have never had a cavity. A dental
graduate student contacts an SRS of 1000 adults and calculates the proportion !̂ in this sample
who have never had a cavity.
a. Identify the mean of the sampling distribution of !̂ .
b. Calculate and interpret the standard deviation of the sampling distribution of !̂ . Check that the
10% condition is met.
c. Is the sampling distribution of !̂ approximately Normal? Check that the Large Counts condition
is met.
d. Find the probability that the random sample of 1000 adults will give a result within 2
percentage points of the true value.
e. If the sample size were 9000 rather than 1000, how would this change the sampling
distribution of !̂ ?
284
7.2 PRACTICE!
The Candy Machine
1. Suppose a VERY large candy machine has 15% orange candies. Imagine taking an SRS of 25 candies from the machine
and observing the sample proportion 𝑝̂ of orange candies.
(a) What is the mean of the sampling distribution of 𝑝̂ ? Why?
(b) Check to see if the 10% condition is met.
(c) Find the standard deviation of the sampling distribution of 𝑝̂ .
(d) Is the sampling distribution of 𝑝̂ approximately Normal? Check to see if the Normal condition is met.
(e) If the sample size were 75 rather than 25, how would this change the sampling distribution of 𝑝̂ ? How would this
impact the Normal condition?
285
Planning for College
2. The superintendent of Miami-Dade County Public Schools wants to know what proportion of middle school students
in his district are planning to attend a four-year college or university. Suppose that 80% of all middle school students in
his district are planning to attend a four-year college or university. What is the probability that a SRS of size 125 will give
a result within 7 percentage points of the true value?
Who owns a Harley?
3. Harley-Davidson motorcycles make up 14% of all the motorcycles registered in the United States. You plan to
interview an SRS of 500 motorcycle owners. How likely is your sample to contain 20% or more who own Harleys?
286
Do Skittles or M&Ms have more orange candies?
Mr. Wilcox believes that Skittles have a higher proportion of
orange candies than M&Ms, while Mrs. Gallas believes the
opposite. Who is correct?
1. Take an SRS of 50 Skittles and an SRS of 50 M&Ms. Calculate the proportion of orange
candies in each sample and find the difference between proportions (Skittles – M&Ms).
Skittles:_______
M&Ms:_______
Difference (Skittles – M&Ms):_______
2. Write the difference on a sticker dot and place on the dot plot at the board. Copy the class
dot plot below.
Difference between proportions (Skittles – M&Ms)
3. What does each dot represent?
4. For the dotplot above, make a prediction about the following:
Shape:
Center (mean):
Variability (SD):
287
A Google search reveals that 21.6% of Skittles are orange and 20% of M&Ms are orange.
5. Describe the sampling distribution of the sample proportion of orange for Skittles (X) and
the sampling distribution of the sample proportion of orange for M&Ms (Y) for samples of
size 50.
Skittles (X)
M&Ms (Y)
Shape:
Mean:
SD:
6. Describe the sampling distribution of the difference between proportions of orange
Skittles and M&Ms (X – Y).
Shape:
Mean of difference between proportions:
Standard deviation of the difference between proportions:
7. Mr. Wilcox and Mrs. Gallas calculated a difference between proportions of 0.08 from their
samples. Calculate the probability of getting this difference in proportions or higher.
288
The Sampling Distribution of 𝑝̂! − 𝑝̂"
Important ideas:
Check Your Understanding
At Westville High School there are 315 seniors and 389 juniors. 65% of the seniors have parking
passes and 42% of the juniors have parking passes. The statistics teacher selects a SRS of 30
seniors and a separate SRS of 30 juniors. Let 𝑝̂! − 𝑝̂" be the difference in the sample proportions of
seniors and juniors that have parking passes.
a. What is the shape of the sampling distribution of 𝑝̂! − 𝑝̂" ? Why?
b. Find the mean of the sampling distribution.
c. Calculate and interpret the standard deviation of the sampling distribution.
d. What is the probability that the difference in sample proportions (senior – junior) of students
with parking passes is greater than 30%?
289
AP Statistics Handout: Lesson 7.2 part 2
Topics: conditions for sampling a proportion - random, 10%, large counts
Guided Notes
Random Condition
The EU Migration Debate and Gender of Immigrants
Vital International
AP Photo/Michael Svarnias
Sympathy for Refugees
Depictions of migrants in the EU
range from sympathetic portrayals
of families escaping violence to
fear-inducing portrayals of groups
of single adult men illegally
crossing borders.
Fear for Jobs & Terrorism
“As you can see from this picture, most of the people coming are young males and,
yes, they may be coming from countries that are not in a very happy state, they may
be coming from places that are poorer than us, but the EU has made a fundamental
error that risks the security of everybody.”
– Nigel Farage, 2016 (U.K.)
Speaking about EU migration: “I said to myself, ‘Wow. They’re all men.’ You look at it.
There are so few women and there are so few children. And not only are they men,
they’re young men. And they’re strong as can be—they’re tough looking cookies. I say,
what’s going on here?”
– Trump, 2015 (during campaign speech)
Trump/Farage statistical claim: 60% of migrants are male
Let’s assume:
1. Trump / Farage’s claim is true (60%) of all EU migrants are male.
2. We collect a sample of 30 recent immigrants and collect data on their gender identity
Under certain conditions:
𝑝̂ ~ Normal(𝜇𝑝̂ = 𝑝, 𝜎𝑝̂ = √
1) Where does the formula come from?
__________________________
𝑝(1−𝑝)
𝑛
)
2) Why is this formula useful?
It gives us a model for ___________________________.
290
Sampling Bias: Why didn’t we analyze U.S. migration data?
Example U.S. immigration data:
Authored by Jill H. Wilson, “Recent Migration to the
United States from Central America: Frequently
Asked Questions.” Congressional Research Service,
Jan 29, 2018: https://fas.org/sgp/crs/row/R45489.pdf
Even though single adult males used to make up over 90% of people apprehended at the border, the
report found that now “the majority of apprehended migrants are families and unaccompanied
children.” (pg. 2)
̂ < 0.50 (proportion who are single adult men)
Found: 𝒑
Problem: Apprehensions are ____________________________________ of all migrants. Single men
may be more successful at crossing the border and are, therefore, undercovered.
Random Sample
𝝁𝒑̂ = 𝒑
Non-random sampling leads to
____________________________!
In this case, using only
apprehensions tends to
______________________ the
true proportion of adult men in the
population.
Non-Random Sample
𝝁𝒑̂ ≠ 𝒑
Condition 1: Random Condition
a) What is the condition?
You must ensure the sample was ___________________________ from the
_______________________ (in the case of an experiment, you must ensure there was random
assignment to treatment)
b) Why must this condition be satisfied?
If condition is not satisfied, your estimator is _________________. If it is satisfied, your sampling
distribution is ___________________ at the true proportion value → 𝜇𝑝̂ = 𝑝
291
10% Condition
Condition 2: 10% Condition
a) What is the condition?
The _______________________ must be less than 10% of the ____________________________.
Formula: n < 0.10(N), where n = sample size & N = population size
b) Why must this condition be satisfied?
It ensures this is true: ___________________
Otherwise, we can’t model the standard deviation of the sampling distribution (in this course).
Intuition behind 10% condition
Situation 1: You are sampling, with replacement, 3 individuals from a population of 10 people. What is
the probability all three individuals identify as male? 60% of the population is male.
Situation 2: You are sampling, without replacement, 3 individuals from a population of 10 people. What
is the probability all three individuals identify as male? 60% of the population is male.
Situation 3: You are sampling, without replacement, 3 individuals from a population of 640 people.
What is the probability all three individuals identify as male? 60% of the population is male.
If n (sample size) is _______________ compared to N (pop. size), process is almost independent
𝑝̂ ~ Normal(𝜇𝑝̂ = 𝑝, 𝜎𝑝̂ = √
This requires __________________ sampling
→ that’s why we need n < 0.10(N)
𝑝(1−𝑝)
)
𝑛
292
Large Counts Condition
Condition 3: Large Counts
a) What is the condition?
The number of “successes” and “failures” must both be greater than or equal to 10.
Formula: ________________ ______________________
b) Why must this condition be satisfied?
This provides evidence that the sampling distribution is ______________________
________________ in shape.
Intuition behind Large Counts
a) n = 11, p = 0.05 → Check the large counts conditions for this situation using the space below:
Extreme p → _______________________
Low sample size → _________________________
Result: skewed distribution (___________________)
b) n = 800, p = 0.05 → Check the large counts conditions for this situation using the space below:
Extreme p → _______________________
High sample size → _________________________
Result: __________________________________
c) n = 21, p = 0.5 → Check the large counts conditions for this situation using the space below:
Non-extreme p → _______________________
Low sample size → _________________________
Result: __________________________________
293
Putting it all together
𝑝̂ ~ Normal(𝜇𝑝̂ = 𝑝, 𝜎𝑝̂ = √
3) Large counts
→ approx. normal
___________
1) Random condition
→ unbiased ___________
𝑝(1−𝑝)
𝑛
)
2) 10% condition
→ calculable __________
Assume the Trump/Farage claim is true: 60% of all EU migrants are male (p = 0.60). If you randomly
sample 30 migrants and calculate the proportion of males (𝑝̂ ):
̂. Check conditions.
a) Describe the sampling distribution of 𝐩
b) Calculate the probability of sampling 15 or fewer males
294
How tall are we?
How tall are high school seniors in Michigan? Attached are the heights of all 50 high school
seniors at a small high school in the upper peninsula.
1. Make a guess at the mean of all 50 students. Make another guess of the standard
deviation of all 50 students.
2. Select a random sample of 5 students and calculate the mean height for the sample.
Repeat for 4 samples total.
Heights:__________________________
Heights:__________________________
Heights:__________________________
Heights:__________________________
!̅
!̅
!̅
!̅
=___________
=___________
=___________
=___________
3. Add your sample means to the dotplot on the board. Sketch it below.
4. Describe the shape, center, and variability of this dotplot.
5. Compare the two dotplots above. How are the dotplots similar? How are they
different?
295
Sample Means
Important ideas:
Check Your Understanding
Every day people watch 1 billion hours of videos on YouTube. That breaks down to every single
person on earth watching YouTube videos for about 8.4 minutes per day. For U.S. teens, in any
given day, the amount of time spent watching YouTube videos is approximately Normal with mean
18.5 minutes and standard deviation 5.3 minutes.
a. Find the probability that a randomly chosen U.S. teen watches YouTube for more than 25
minutes in a given day.
Suppose we choose an SRS of 10 U.S. teens. Let x = the mean amount of time spent watching
YouTube videos for the sample.
b. What is the mean of the sampling distribution of x ?
c. Calculate and interpret the standard deviation of the sampling distribution of x . Verify that
the 10% condition is met.
d. Find the probability that the mean amount of time spent watching YouTube for the teens
in the sample exceeds 25 minutes.
296
AP Statistics Handout: Lesson 7.3
Topics: µ vs. 𝑥̅, Sampling distribution for a mean, central limit theorem and conditions
Guided Notes
µ vs. 𝑥̅
In your own words, describe the purpose of SNAP benefits:
SNAP benefits should cover the following essential monthly bundle of groceries for one person:
• 4 boxes of cereal, 1 gallon of milk, 5 banana bunches, 2 gallons of orange juice
• 3 loaves of bread, 4 packages of sliced turkey (9-oz each), 3 packages of sliced provolone (10
count), 4 packages of romaine lettuce (7 oz each), 12 tomatoes, small mayo jar (15 oz), 30
apples, 4 bags of mixed nuts (8 oz each), 5 diet coke 6-packs.
• 15 lbs boneless chicken breasts, 6 packs of frozen mixed veggies (16 oz each), 4lbs white rice, 10
cans of black beans
• Salt (5 oz), pepper (5 oz), olive oil (17 oz)
We’ll call this grocery list the essentials bundle. We will use sampling to determine if SNAP benefits
cover the average (mean) cost of this grocery bundle in U.S. grocery stores.
µ = ___________________ mean
• Parameter
𝑥̅ = _________________ mean
• Statistic
• Estimator of µ
How much SNAP funding would it take to support an essentials-only diet? We obtain an SRS of 31 U.S.
grocery stores, find the cost for the monthly essentials bundle at each store, and calculate the mean
cost among the sample: $201.24
Population:
Parameter:
Sample:
Statistic:
297
Sampling Distribution for a Mean
Welcome to Theory Land – “The not real place.”
In fiscal year 2020, the maximum monthly SNAP allotment* for a household of one person (who could
not earn income) was $194 in the U.S. Let’s assume this is the true mean cost of the essentials bundle in
the United States.
*This is the max federal allotment for the continental U.S. (not Alaska/Hawaii). Source: U.S. Department of Agriculture, “SNAP Fiscal Year 2020
Cost-of-Living Adjustments,” https://fns-prod.azureedge.net/sites/default/files/media/file/COLA%20Memo%20FY%202020.pdf
Data
µ = $194
Prices for bundle
at all stores
Realm of Perfect Parameters
Population Distribution
Data
Realm of Sucky Statistics
Prices for bundle
at 16 sampled
stores (n = 16)
Means
̅
𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
Sample Distribution
Sampling Distribution
Means from each
sample
Screenshots taken from sample means applet at Rice Virtual Lab in Statistics (onlinestatbook.com):
http://onlinestatbook.com/stat_sim/sampling_dist/index.html
Sampling Distribution for a Mean - the formula
Under certain conditions:
𝑥̅ ~ 𝑁𝑜𝑟𝑚 (𝜇𝑥̅ = 𝜇, 𝜎𝑥̅ =
We are centered at the true population
mean, meaning our estimates are
___________________.
Notation:
𝜇𝑥̅ → center of sampling dist. of 𝑥̅
𝜇 → population mean
𝜎
√𝑛
)
We _____________________, so spread
affected by sample size. Higher sample
size means _______________________.
𝜎𝑥̅ → standard deviation of the sampling dist. of 𝑥̅
𝜎 → standard deviation of population values
298
Spread and Sample Size
µ = $194
Population Distribution
of Bundle Costs
Assume the standard deviation of bundle
costs among the population (σ) is $26:
Calculate the standard deviation of the
sampling distribution (𝜎𝑥̅ )…
̅
𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
̅ vc
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
𝒙
̅vc𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
̅ vc
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
𝒙
̅𝒙
̅ v𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
Sampling Distribution
of Sample Means
(n = 25)
a) When n = 25
Sampling Distribution
of Sample Means
(n = 5)
b) When n = 5
v
v
Since we divide by sample size (n), spread ____________________ as sample size _________________.
Full sampling distribution calculation: SNAP benefits example
If the monthly amount SNAP provides ($194) is the exact true mean cost of the essentials bundle in the
U.S., what is the probability that a random sample of 31 grocery stores finds an average essentials
bundle cost of $201.24 or more? Assume the population standard deviation of bundle prices is $26 and
all conditions are met:
299
Central Limit Theorem and Conditions
Condition 1: Random Condition
a) What is the condition?
You must ensure the sample was ___________________________ from the
_______________________ (in the case of an experiment, you must ensure there was random
assignment to treatment)
b) Why must this condition be satisfied?
If condition is not satisfied, your estimator is _________________. If it is satisfied, your sampling
distribution is ___________________ at the true population mean value → 𝜇𝑥̅ = 𝜇
µ = $194
Population Distribution
of grocery bundle costs
̅
𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
̅𝒙
̅𝒙
̅𝒙
̅𝒙
̅
𝒙
What if the SNAP program only sampled
conveniently-located grocery stores near its
offices in downtown San Francisco?
Sampling distribution from random samples
µ𝑥̅ = ___
Sampling distribution from convenience
samples (only stores in San Francisco)
µ𝑥̅ ≠ ___
When we don’t randomly sample, we get __________________________. In this case, we tended to
overestimate the mean price of the groceries (cost of living in San Francisco is high).
Condition 2: 10% Condition
a) What is the condition?
The _______________________ (n) must be less than 10% of the ____________________________ (N).
Formula: n < 0.10(N)
b) Why must this condition be satisfied?
It ensures this is true: ___________________
Otherwise, we can’t model the standard deviation of the sampling distribution (in this course).
300
Condition 3: Normal/Large Sample
a) What is the condition?
i) The _____________________ distribution must be normal _____
ii) The sample size is ___________________ (n ≥ 30)
b) Why must this condition be satisfied?
This provides evidence that the sampling distribution is __________________________ in shape.
Normal Population
Non-Normal Population
Sampling distribution
is approximately
normal (___________
________________)
Sampling distribution
is approximately
normal only for ____
__________________
Central limit theorem (CLT): when n is large, the sampling distribution for 𝑥̅ is ______________________
________________________
• Even with a highly skewed population, the sampling distribution is close to normal at n = 25. So
we make the rule ___________ just to be safe.
• Useful because we ______________________ the population’s shape! No matter the
population shape, we can model the _______________________________________________,
if there’s a high enough sample size (n).
All the conditions summarized:
𝑥̅ ~ 𝑁𝑜𝑟𝑚 (𝜇𝑥̅ = 𝜇, 𝜎𝑥̅ =
3) Normal/Large Sample
→ approx. normal
___________ (by CLT)
1) Random condition
→ unbiased ___________
301
𝜎
√𝑛
)
2) 10% condition
→ calculable __________
Assume the monthly amount SNAP provides ($194) is the true mean cost of the essentials bundle in U.S.
grocery stores. Also assume the population standard deviation of bundle prices is $26. If you obtain a
random sample of 31 grocery stores and calculate their average price (𝑥̅ ) for the essentials bundle…
̅. Check conditions.
a) Describe the sampling distribution of 𝒙
b) Calculate the probability of obtaining a sample mean bundle price of $201.24 or higher:
Discussion
Discussion Question: A real random sample of 31 U.S. grocery stores found the average cost of the
essentials bundle was $199.45. If we assume the true average costs for this bundle in American grocery
stores is $194 (the max allotment provided by SNAP), what is the probability of finding a sample mean of
$199.45 or higher from 31 stores? Does this probability make you doubt that the assumption that the
SNAP allotment covers the true mean price of the essential groceries? Why or why not?
302
Who has better ACT scores?
The ACT test is scored with whole numbers from 0 to 36. We will use the applet at
http://onlinestatbook.com/stat_sim/sampling_dist/ to take samples of ACT scores from EK and
Rockford.
Click “Begin” and you will see the population distribution of ACT scores from EK.
1. Describe the shape, center, and variability of the distribution of ACT scores for EK.
2. Click “Animated” to take a sample of 5 ACT scores.
List 5 estimated scores here:________________ Estimated mean (blue box):________
Click “Animated” several more times. Then click “10,000” to take 10,000 samples of size 5.
3. The blue boxes make the sampling distribution of x . How do we know that the
sampling distribution of x is approximately normal (hint: see previous lesson)?
4. Now let’s look at the distribution of ACT scores for Rockford. Click “Clear lower 3”
and then change the distribution from “Normal” to “Skewed”. What is the shape of
this distribution? Why does this distribution make sense for our archrival Rockford?
Change both of the bottom two dropdown menus to “Mean”. The first one should be “N=2”
and the second one should be “N=25”. The click “10,000” to take 10,000 samples.
5. Describe the shape of the sampling distribution of x when N = 2.
6. Describe the shape of the sampling distribution of x when N = 25.
303
The Central Limit Theorem
Important ideas:
Check Your Understanding
Among iPhone users who share their data with Apple, the weekly screen time is skewed to the
right with µ = 13.5 hours and σ = 3.75 hours. A random sample of 100 iPhone users are selected
and the mean weekly screen time (𝑥̅ ) of the sample is calculated.
a. Describe the shape of the sampling distribution of 𝑥̅ for samples of 100 randomly selected
iPhone users. Justify your answer.
b. Find the mean and standard deviation of the sampling distribution of 𝑥̅ . Be sure to check
the 10% condition.
c. Calculate the probability that the weekly screen time for the sample is between 12 and 13
hours.
304
ACT Scores: Which school is better?
ACT scores at Ardrey Kell High School are Normally distributed
with mean 26 and standard deviation 3. ACT scores at Providence
HS are skewed to the right with mean 25 and standard deviation 5.
1. We randomly select 25 students from AKHS and 30 students from PHS. Use the
information given to describe the sampling distributions of the average ACT scores for the
two samples.
AKHS
PHS
Shape:
Mean:
SD:
2. Suppose we took a sample of 25 students from AKHS and a sample of 30 students from
PHS and found the difference in the sample means. Describe the sampling
distribution of the difference in mean ACT scores (AKHS – PHS).
Mean:
Standard Deviation:
Shape:
3. Calculate the probability that random sample of 25 AKHS students has a higher mean
ACT score than the random sample of 30 PHS students.
305
The Sampling Distribution of !̅! − !̅"
Important ideas:
Check Your Understanding
Is it more expensive to live in California or Florida? Monthly cost of living expenses for California are
approximately Normal with mean µC = $10,800 and standard deviation s C = $3,200. Monthly cost of
living expenses for Florida are approximately Normal with mean µF = $8,500 and standard
deviation s F = $2,700. Suppose we select independent SRSs of 16 residents of California and 9
residents of Florida and calculate the sample mean monthly cost of living, xC and xF .
a. What is the shape of the sampling distribution of xC - xF ? Why?
b. Find the mean and standard deviation of the sampling distribution of xC - xF .
c. Calculate the probability that the average monthly cost of living expense of the 16 randomly
selected California residents is less than the average monthly cost of living expense of the 9
randomly selected Florida residents.
306
Sampling Distributions for Sample Proportions
One Sample
Two Samples
Parameter
Statistic
Mean of Sampling
Distribution
Standard Deviation
of Sampling
Distribution
What condition must
be met to use the
standard deviation
formula?
When is the
sampling distribution
approximately
normal?
z-score formula
307
Sampling Distributions for Sample Means
One Sample
Two Samples
Parameter
Statistic
Mean of Sampling
Distribution
Standard Deviation
of Sampling
Distribution
What condition must
be met to use the
standard deviation
formula?
When is the
sampling distribution
approximately
normal?
z-score formula
308
CENTRAL LIMIT THEOREM PENNIES LAB
In this lab we will be interested in the age of the pennies (in years) and not the year the pennies
were produced. To find the age of a penny, simply subtract the year on the penny from the current
year.
1. Make a dotplot of the ages of the 25 pennies below.
8
7
6
5
4
3
2
1
0 <----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o >
0 2 4
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
This is the population distribution. Describe the distribution:
_______________________________________________________________________________
2. Now take a random sample of three pennies. Calculate the average age of the sample. This is
called a sample mean. Record your sample mean: ______
3. Replace the three pennies. Then take another sample of three pennies. Find the sample mean and
record it. ______
4. Do this a total of five times. Record your three other sample means below:
_______
_______
______
309
5. Write each of your sample means on a different post-it note and then take your post-it note to the
histogram on the whiteboard. Record the results of the histogram on the whiteboard below:
10
9
8
7
6
5
4
3
2
1
0 <----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o >
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
6. We will now change the size of the sample. Take a random sample of 10 pennies. Calculate the
average age of the pennies. Record it here: ________
7. Do this a total of five times. Record your four other sample means below:
_______
_______
_______
_______
8. Write each of your sample means on a different post-it note and then take your post-it note to the
histogram on the whiteboard. Record the results of the histogram on the whiteboard below:
20
18
16
14
12
10
8
6
4
2
0 <----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o >
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
Summary:
•
If the population distribution is normal, the sampling distribution of x is also
________________. This is true for any sample size n.
•
If the population distribution is NOT normal, the Central Limit Theorem (CLT)
tells us that the sampling distribution of x will be approximately normal in most
cases if ____________.
310
7.3 PRACTICE!
Buy Me Some Peanuts and Sample Means
1. At the P. Nutty Peanut Company, dry roasted, shelled peanuts are placed in jars by a machine. The distribution of
weights in the bottles is approximately Normal, with a mean of 16.1 ounces and a standard deviation of 0.15 ounces.
(a) Without doing any calculations, explain which outcome is more likely, randomly selecting a single jar and finding the
contents to weigh less than 16 ounces or randomly selecting 10 jars and finding the average contents to weigh less than
16 ounces.
(b) Find the probability of each event described above. Since the distribution is normal you can use “normalcdf” on your
calculator.
Single jar weighing 16 oz or less:
10 jars weighing 16 oz or less:
311
Mean Texts
2. Suppose that the number of texts sent during a typical day by a randomly selected high school student follows a rightskewed distribution with a mean of 15 and a standard deviation of 35. Assuming that students at your school are typical
texters, how likely is it that a random sample of 50 students will have sent more than a total of 1000 texts in the last 24
hours?
Bad carpet
3. The number of flaws per square yard in a type of carpet material varies with mean 1.6 flaws per square yard and
standard deviation 1.2 flaws per square yard. The population distribution cannot be Normal, because a count takes only
whole-number values. An inspector studies 200 square yards of the material, records the number of flaws found in each
square yard, and calculates X, the mean number of flaws per square yard inspected. Find the probability that the mean
number of flaws exceeds 2 per square yard.
312
Estimating a Population Proportions
Important ideas:
Check Your Understanding
A community activist group in Austin, Texas wanted a particular issue to be placed on the ballot of the
upcoming election. To make it on the ballot, 20,000 valid signatures were needed. The group turned
in their petition with 24,598 signatures. To pass the validity test 20,000/24,598 = 81.3% of the
signatures must be valid. It is too time consuming to check all of the signatures, so a random sample
of signatures are checked. The individual checking the signatures needs to be 95% confident that the
true proportion of valid signatures are estimated with, at most, a 2% margin of error.
1. Using a conservative estimate for 𝑝̂, how large of a sample is needed?
2. In the activist group’s previous petition, 85% of the signatures were valid. Using this value as
a guess for 𝑝̂, find the sample size needed for a margin of error of at most 2 percentage
points with 95% confidence. How does this compare with the required sample size from
Question 1?
3. What if the company president demands 99% confidence instead of 95% confidence? Would
this require a smaller or larger sample size, assuming everything else remains the same?
Explain your answer.
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
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®
2014 AP STATISTICS FREE-RESPONSE QUESTIONS
3. Schools in a certain state receive funding based on the number of students who attend the school. To
determine the number of students who attend a school, one school day is selected at random and the number
of students in attendance that day is counted and used for funding purposes. The daily number of absences at
High School A in the state is approximately normally distributed with mean of 120 students and
standard deviation of 10.5 students.
(a) If more than 140 students are absent on the day the attendance count is taken for funding purposes, the
school will lose some of its state funding in the subsequent year. Approximately what is the probability
that High School A will lose some state funding?
(b) The principals' association in the state suggests that instead of choosing one day at random, the state should
choose 3 days at random. With the suggested plan, High School A would lose some of its state funding in
the subsequent year if the mean number of students absent for the 3 days is greater than 140. Would High
School A be more likely, less likely, or equally likely to lose funding using the suggested plan compared
to the plan described in part (a)? Justify your choice.
(c) A typical school week consists of the days Monday, Tuesday, Wednesday, Thursday, and Friday. The
principal at High School A believes that the number of absences tends to be greater on Mondays and
Fridays, and there is concern that the school will lose state funding if the attendance count occurs on
a Monday or Friday. If one school day is chosen at random from each of 3 typical school weeks, what
is the probability that none of the 3 days chosen is a Tuesday, Wednesday, or Thursday?
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®
2015 AP STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
PartB
Question 6
Spend about 25 minutes on this part of the exam.
Percent of Section II score-25
Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
6. Com tortillas are made at a large facility that produces 100,000 tortillas per day on each of its two production
lines. The distribution of the diameters of the tortillas produced on production line A is approximately normal
with mean 5.9 inches, and the distribution of the diameters of the tortillas produced on production line B is
approximately normal with mean 6.1 inches. The figure below shows the distributions of diameters for the
two production lines.
Production Line A
Production Line B
I
I
I
I
I
I
I
I
I
I
I
I
,,....\
I
I
I
5.7
5.8
6.0
5.9
6.1
Diam t r (inches)
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
''
6.2
6.3
The tortillas produced at the factory are advertised as having a diameter of 6 inches. For the purpose of quality
control, a sample of 200 tortillas is selected and the diameters are measured. From the sample of 200 tortillas, the
manager of the facility wants to estimate the mean diameter, in inches, of the 200,000 tortillas produced on a
given day. Two sampling methods have been proposed.
Method 1: Take a random sample of 200 tortillas from the 200,000 tortillas produced on a given day.
Measure the diameter of each selected tortilla.
Method 2: Randomly select one of the two production lines on a given day. Take a random sample of
200 tortillas from the 100,000 tortillas produced by the selected production line. Measure the diameter of
each selected tortilla.
(a) Will a sample obtained using Method 2 be representative of the population of all tortillas made that day,
with respect to the diameters of the tortillas? Explain why or why not.
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®
2015 AP STATISTICS FREE-RESPONSE QUESTIONS
(b) The figure below is a histogram of 200 diameters obtained by using one of the two sampling methods
described. Considering the shape of the histogram, explain which method, Method 1 or Method 2, was most
likely used to obtain a such a sample.
25
(/)
Cd
20
15
t
10
5
0
5.775
5.850
5.925 6.000 6.075
Diameter (inche )
6.150
6.225
(c) Which of the two sampling methods, Method 1 or Method 2, will result in less variability in the diameters of
the 200 tortillas in the sample on a given day? Explain.
Each day, the distribution of the 200,000 tortillas made that day has mean diameter 6 inches with standard
deviation 0.11 inch.
(d) For samples of size 200 taken from one day's production, describe the sampling distribution of the sample
mean diameter for samples that are obtained using Method 1.
(e) Suppose that one of the two sampling methods will be selected and used every day for one year (365 days).
The sample mean of the 200 diameters will be recorded each day. Which of the two methods will result in
less variability in the distribution of the 365 sample means? Explain.
(f) A government inspector will visit the facility on June 22 to observe the sampling and to determine if the
factory is in compliance with the advertised mean diameter of 6 inches. The manager knows that, with both
sampling methods, the sample mean is an unbiased estimator of the population mean. However, the manager
is unsure which method is more likely to produce a sample mean that is close to 6 inches on the day of
sampling. Based on your previous answers, which of the two sampling methods, Method 1 or Method 2, is
more likely to produce a sample mean close to 6 inches? Explain.
316
Unit 6
Chapter 8
Estimating
Proportions with
Confidence
317
AP Statistics Handout: Lesson 8.1
Topics: estimation, margin of error, interpreting confidence intervals and confidence levels
Guided Notes
Estimation and margin of error
AP Photos
Getty Images
Hurricane Katrina – Sheltering at the Superdome
FEMA (Federal Emergency Management Agency) is trying to determine the amount of relief aid (medical
supplies, beds, food, etc.) to provide to the SuperDome. You are FEMA’s head statistician, and you are
tasked with estimating the number of people sheltering at the SuperDome.
Cluster Random Sample
-Unbiased
• If sections equal size
-Saves time
• Avoids counting everyone
• Stadium already in sections
Sampled section 1: 152 people
Sampled section 2: 122 people
Sampled section 3: 192 people
Sampled section 4: 84 people
Sampled section 5: 145 people
Sampled section 6: 202 people
Sampled section 7: 96 people
Sampled section 8: 104 people
Sampled section 9: 156 people
Sampled section 10: 124 people
Average number of people per section:
𝑥̅ = 137.7
Number of seating sections in stadium:
120 sections
Estimate for total number of people:
137.7 ∗ 120 = _______________
Is this the exact number of people sheltering in the SuperDome?
____ → Even though our estimator should be unbiased, there is
___________________________ in our random sample. We may
overestimate or underestimate the true population size by chance.
318
Confidence Interval: point estimate ± margin of error
___________________
16,524 people
______________________
16,524 people
Narrower intervals are _______________________, but they make you _______________________
you’ve captured the true parameter value.
16,524 people
Wider intervals give _____________________ you’ve captured the true parameter, but they’re often
not as ______________.
Constructing and interpreting a confidence interval
138 is probably not the exact true mean number of people per section. What confidence interval should
you give so that you capture the true mean?
Imagine the sampling distribution for 𝑥̅ is normal with 𝜇𝑥̅ = 138, 𝜎𝑥̅ = 3. Recall the 68-95-99.7 rule:
68% of all possible sample means are
within 1𝝈𝒙̅ of the true mean
95% of all possible sample means are
within 2𝝈𝒙̅ of the true mean
319
Interval: __________________
Interpretation:
We are _______ confident the interval from 132 to 144 _________________ the true mean number of
people per section.
Wrong Interpretation: There is a 95% probability that the interval from 132 to 144 captures the true
mean number of people per section.
• Why not “probability”?
If it _______ in the interval, the probability
it’s in the interval is _________.
If it’s _______ in the interval, the probability
it’s in the interval is _________.
The parameter has a fixed value. It’s already in the interval or it’s not.
Instead, we say we are 95% _________________ we captured the true mean. Even though our interval
already did/didn’t capture the true value, we (as ignorant humans) _____________________. So, we
can only be certain levels of ‘confident’ that we captured the true value.
320
Interpreting a confidence level
Confidence Level:
95%
Sample Size (n):
20
Confidence Level:
If you raise the __________________
____________, the confidence interval
becomes _____________.
99.7%
Sample Size (n):
20
You have higher confidence of capturing
the true value, but your interval is
_________________________.
Confidence Level:
If you raise the ____________________,
the confidence interval becomes
_________________.
95%
Sample Size (n):
80
You have more precise interval, even at
the __________ confidence level.
321
Applet (confidence levels)
http://digitalfirst.bfwpub.com/stats_applet/stats_applet_4_ci.html
Source: Digital First project from Bedford, Freeman, & Worth publishers
Interpret a 95% confidence level:
“If we take many samples of the
______________ from this population, about
95% of them will result in an interval that
_______________ the actual parameter value.”
Discussion
Discussion Question: Imagine you calculated the following three confidence intervals for the number of
people taking shelter in the Superdome:
a) Which interval would you report to FEMA?
Why?
b) How would you explain to FEMA what the interval means?
322
Guess the Mystery Proportion
Mrs. Mason and Goodenough wants to implement a new reward system. When a student
does somethinggreat, they randomly pick a bead from the bead jar. If the bead is red,
they get a piece ofcandy. If not, they don’t win a prize. The question is, what are the
chances that a studentchooses a red bead? Each group will get a sample of beads and
will create a confidence interval to estimate the true proportion of red beads. The group
with the smallest intervalthat captures the true proportion wins a prize!
1. You will select a random sample of 20 beads from the jar. Calculate the proportion of
beads that are red (write this as a decimal).
Proportion red:_______. This is your point estimate for the true proportion of red beads.
2. Identify the population, parameter, sample, and statistic.
Population:______________________________
Parameter:____________________
Sample:________________________________
Statistic:____________________
3. Now you are going to change your point estimate into an interval of values by adding and
subtracting some value from your point estimate (the number you add and subtract is called
your margin of error). You can choose any amount to add and subtract, but remember, the
smallest interval that captures the truth is the winner. What margin of error do you want to
use? Why?
4. Use your point estimate and chosen margin of error to write an interval that you think
contains the true proportion of red beads.
5. How confident do you feel that your interval captures the true proportion? Answer with a
percentage.
6. One of the groups got (0.27, 0.33) as their interval. What was their point estimate? What
was their margin of error?
7. One group claims that the true proportion of red beads 0.25. Does your interval support or
deny this claim? Why?
323
The Idea of a Confidence Interval
Important ideas:
Check Your Understanding
A random sample of 100 adults are asked if they pay for monthly subscriptions that they do not use,
like a magazine, app, or online program simply. Many do, because they have never taken the time
to cancel the subscription. A 95% confidence interval for the proportion of adults who to pay for
subscriptions they do not use is 0.352 to 0.548.
a. Interpret the confidence interval.
b. Calculate the point estimate and the margin of error.
c. Based upon this survey, a reporter claims that a majority of adults continue to pay for monthly
subscriptions they do not use. Use the confidence interval to evaluate this claim.
324
8.1 PRACTICE!
Losing Weight
A Gallup Poll in November 2014 found that 59% of the people in its sample said “Yes” when asked,
“Would you like to lose weight?” Gallup announced: “For results based on the total sample of national
adults, one can say with 95% confidence that the margin of (sampling) error is ±3 percentage points.”
(a) Explain what the margin of error means in this setting.
(b) State and interpret the 95% confidence interval.
(c) Interpret the confidence level.
The admissions director from Florida International University found that (107.8, 116.2) is a 95%
confidence interval for the mean IQ score of all freshmen. Comment on whether or not each of the
following explanations is correct.
(a) There is a 95% probability that the interval from 107.8 to 116.2 contains μ.
325
(b) There is a 95% chance that the interval (107.8, 116.2) contains X.
(c) This interval was constructed using a method that produces intervals that capture the true mean in
95% of all possible samples.
(d) 95% of all possible samples will contain the interval (107.8, 116.2).
(e) The probability that the interval (107.8, 116.2) captures μ is either 0 or 1, but we don’t know which.
How do Confidence Intervals Change…
(a) How does the shape of the confidence interval change if the confidence level increases from 90% to
95%?
(b) How would the shape of a confidence change if the sample size was decreased? Assume the new,
smaller sample size still meets all of the normality conditions.
326
What does “95% confident” mean?
In this Activity, you will use the applet at www.tinyurl.com/appletCI to learn what it means to
say we are “95% confident” that our confidence interval captures the true proportion.
1. Use the Confidence Intervals for Proportions applet. Set the population proportion to
0.5, the confidence level to 95% and the sample size to 75.
2. Click “Sample” to choose an SRS and display the resulting confidence interval. The
confidence interval is displayed as a horizontal line segment with a dot representing
the sample proportion in the middle of the interval. The true proportion (p) is the green
vertical line.
Did the first confidence interval capture the
true proportion?
Repeat this 10 times and sketch what you
see to the right. How many of the intervals
capture the true proportion?
3. “Reset” and then take a total of 100 confidence intervals (sample 25 four times). How
many out of 100 captured the true proportion? Is this surprising? Why?
4. Watch your confidence intervals as you drag the confidence level from 95% to 99%
(don’t “Reset). What happens to the intervals when the confidence level is increased?
Why does this make sense?
5. “Reset”, then sample 100 times at an 80% confidence interval. What percent of the
intervals capture the true proportion?
Interpret the confidence level:
6. Now we will see what happens when we adjust the sample size. Change the sample
to 20 and sample for 1 interval. Then change it to 250 and sample for 1 interval.
What happens to the interval when the sample size is increased? Why?
327
Interpreting the Confidence Level
Important ideas:
Check Your Understanding
The gym teacher of a large high school wants to estimate the mean number of pushups students at
this school can do in one minute. He selects a random sample of 30 students from those who are
there after school for sports practices. He records how many pushups each of the students in the
sample can do in one minute. He determines that he is 90% confident that the interval from 24.1 to
28.5 captures the mean number of pushups that students at this school can do in one minute.
a. Interpret the confidence level.
b. Explain what would happen to the length of the interval if the confidence level were increased
to 99%.
c. How would a 90% confidence interval based on a sample of size 200 compare to the original
90% interval?
d. Describe one potential source of bias in this study that is not accounted for by the margin of
error.
328
AP Statistics Handout: Lesson 8.2
Topics: review sampling distribution for 𝑝̂ , 95% confidence interval for 𝑝̂ , four step process
Guided Notes
Review: Sampling Distribution for 𝑝̂
Vital International
AP Photo/Michael Svarnias
Sympathy for Refugees
Depictions of migrants in the EU
range from sympathetic portrayals
of families escaping violence to
fear-inducing portrayals of groups
of single adult men illegally
crossing borders.
Fear for Security
“As you can see from this picture, most of the people coming are young males and,
yes, they may be coming from countries that are not in a very happy state, they may
be coming from places that are poorer than us, but the EU has made a fundamental
error that risks the security of everybody.”
– Nigel Farage, 2016 (U.K.)
Speaking on EU migration: “I said to myself, ‘Wow. They’re all men.’ You look at it.
There are so few women and there are so few children. And not only are they men,
they’re young men. And they’re strong as can be—they’re tough looking cookies. I say,
what’s going on here?”
– Trump, during 2015 campaign speech
Trump/Farage statistical claim (conservatively): 60% of migrants are male
Let’s assume:
1. Trump / Farage’s claim is true (60%) of all EU migrants are male.
2. We collect a sample of 30 recent immigrants and collect data on their gender identity
We ________________ the true
proportion by 0.10
In most real-life scenarios:
-You _______________________ the true proportion.
-You survey a random sample of 30 immigrants. 15 of them are male.
So, you calculate the proportion of sample who are male:
15
- 𝑝̂ = = ___________
30
- Your estimate for the true proportion (p) is the sample proportion (𝑝̂ )
329
How do we determine how far off our estimates will usually be? We use the sampling distribution.
Under certain conditions:
𝑝̂ ~ Normal(𝜇𝑝̂ = 𝑝, 𝜎𝑝̂ = √
Centered at true proportion (______________)
𝑝(1−𝑝)
𝑛
)
____________________________ of our
estimates from true proportion value
Calculations:
𝑝̂ ~ Normal(𝜇𝑝̂ = 𝑝, 𝜎𝑝̂ = √
𝑝(1−𝑝)
)
𝑛
𝑝̂ ~ Normal(𝜇𝑝̂ = .6, 𝜎𝑝̂ = √
.6(.4)
)
30
𝑝̂ ~ Norm(𝜇𝑝̂ = .6, 𝜎𝑝̂ = .089)
Typical distance of estimates
from true proportion value.
________________________
by .089
We ________________ the true
proportion by 0.10
a) Was our estimate’s error amount (0.10) pretty typical? Justify using the sampling distribution above.
There should be a way to report both:
a) Our estimate
b) How far off this estimate might be
One way to do this is with a _________________________________!
330
95% Confidence Interval for 𝑝̂
The EU Migration Study – Spain
We will use data from Spain, which has the best records on EU migrants (both documented and
undocumented migrants). According to a report* in Comparative Migration Studies, Spain has a “nationwide rolling local population register, which provides a comprehensive register of the population, also
including unauthorized immigrants.” The Immigrant Citizens Survey** conducted a random sample of
this database, surveying recent immigrants to Spain who were 15 and older.
Recall:
𝑝 = ___________________ proportion
• Parameter
𝑝̂ = _________________ proportion
• Statistic
• Estimator of p
Scenario (Mock Data): In reality, we don’t know the true proportion of recent immigrants to Madrid
(Spain’s immigration hub) who are male because it’s almost impossible to collect records about each
individual. So, instead, imagine we surveyed a random sample of 34 immigrants in Madrid’s address
registries. Of the random sample, 15 identified as male. Estimate the true proportion of Madrid’s recent
immigrants who identify as male, with 95% confidence.
Since we don’t know the true proportion (p),
we use the sample proportion (𝑝̂ ) to generate
our sampling distribution:
Under certain conditions:
𝑝(1−𝑝)
𝑝̂ ~ Normal(𝜇𝑝̂ = 𝑝, 𝜎𝑝̂ = √
Consider all possible
random samples that could
have happened (sampling
distribution)
𝑛
̂ , 𝑆𝐸𝑝̂ = √
𝑝̂ ~ Normal(𝜇𝑝̂ = 𝒑
)
̂ (1−𝒑
̂)
𝒑
𝑛
)
.441 (1−.441 )
)
34
𝑝̂ ~ Normal(𝜇𝑝̂ = .441 , 𝑆𝐸𝑝̂ = √
𝑝̂ ~ Normal(𝜇𝑝̂ = . 441, 𝑆𝐸𝑝̂ = . 𝟎𝟖𝟓)
Confidence Interval:
𝑝̂ ± 1.96 ∗ 𝑆𝐸𝑝̂
= .441 ± 1.96 ∗ (. 𝟎𝟖𝟓)
= .441 ± 1.67 = (0.274, 0.608)
Side note: Why do we use 1.96 in the formula?
Interpretation:
We are 95% confident the interval from 27.4% to 60.8% ________________ the ___________________
of Madrid immigrants who are male.
*Reichel and Morales, “Surveying immigrants without sampling frames – evaluating the success of alternative field methods,” Comparative
Migration Studies (2017) 5:1 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5209407/pdf/40878_2016_Article_44.pdf
**Huddleston, Thomas and Dag Tjaden, Jasper. ”Immigrant Citizens Survey: How immigrants experience integration in 15 European cities,” King
Baudouin Foundation and The Migration Policy Group. May 2012.
331
Formula for 95% confidence interval for one proportion:
𝑝̂(1−𝑝̂)
𝑝̂ ± 1.96 ∗ 𝑆𝐸𝑝̂ →
𝑝̂ ± 1.96 ∗ √
𝑛
Imagine we surveyed a random sample of 34 immigrants in Madrid’s address registries. Of the random
sample, 15 identified as male.
a) Construct and interpret a 95% confidence interval for the true proportion of Madrid’s recent
immigrants who identify as male. (use the four step process)
b) Use your interval to evaluate the validity of Trump/Farage’s claim that 60% of immigrants are male.
332
Which way will the Hershey Kiss land?
When you toss a Hershey Kiss, it sometimes lands flat and sometimes lands on its side.
What proportion of tosses will land flat?
Each group of four selects a random sample of 50 Hershey’s Kisses to bring back to their desks.
Toss the 50 Kisses and then calculate the proportion that land flat. Let p̂ = the proportion of the
Kisses that land flat.
1. What is your point estimate for the true proportion that land flat? ________
2. Identify the population, parameter, sample and statistic.
Population:
Parameter:
Sample:
Statistic:
3. Was the sample a random sample? Why is this important?
4. What is the formula for calculating the standard deviation of the sampling distribution of p̂ ?
5. What condition must be met to use this formula? Has it been met?
6. We don’t know the value of p (that’s the whole point of a confidence interval) so we will use p̂
instead. Calculate the standard deviation.
7. Would it be appropriate to use a normal distribution to model the sampling distribution of p̂ ?
Justify your answer.
333
Name:
Hour:
Date:
8. In a normal distribution, 95% of the data lies within ______ standard deviations of the mean.
This value is called the critical value. Use table A or InverseNorm to find these critical
values:
80% of the data lies within ________ standard deviations of the mean
90% of the data lies within ________ standard deviations of the mean
95% of the data lies within ________ standard deviations of the mean
99% of the data lies within ________ standard deviations of the mean
9. Calculate the margin of error for a 95% interval by multiplying the critical value and standard
deviation you found. Show your work.
10. Find the 95% confidence interval using point estimate +/- margin of error.
11. Add your interval to the graph on the board. Sketch the graph below.
12. What do you think is the true proportion of kisses that land flat is?
334
Name:
Hour:
Date:
Constructing a Confidence Interval for p
Important ideas:
Check Your Understanding
What do you want to be when you grow up? According a nationwide survey of a random sample of
1000 kids under the age of 12, some kids want to be a ninja, a dragon keeper, a dancing unicorn,
and even an octopus. Fifty-five of the 1000 kids want to be a doctor. We would like to use this study
to find a 98% confidence interval for the true proportion of kids who want to be a doctor.
a. Identify the parameter of interest.
b. Check if the conditions for constructing a confidence interval for p are met.
c. Find the critical value for a 98% confidence interval. Then calculate the interval.
d. Interpret the interval in context.
335
AP Statistics Handout: Lesson 8.2 part 2
Topics: critical values, confidence interval calculator steps, sample size calculations
Guided Notes
Vital International
AP Photo/Michael Svarnias
Sympathy for Refugees
Depictions of migrants in the EU
range from sympathetic portrayals
of families escaping violence to
fear-inducing portrayals of groups
of single adult men illegally
crossing borders.
Fear for Security
The EU Migration Study – Spain
We will use data from Spain, which has the best records on EU migrants (both documented and
undocumented migrants). According to a report* in Comparative Migration Studies, Spain has a “nationwide rolling local population register, which provides a comprehensive register of the population, also
including unauthorized immigrants.” The Immigrant Citizens Survey** conducted a random sample of
this database, surveying recent immigrants to Spain who were 15 and older.
95% confidence interval (with mock data): Imagine we surveyed a random sample of 34 immigrants in
Madrid’s address registries. Of the random sample, 15 identified as male. Estimate the true proportion
of Madrid’s recent immigrants who identify as male, with 95% confidence.
̂ , 𝑆𝐸𝑝̂ = √
𝑝̂ ~ Normal(𝜇𝑝̂ = 𝒑
𝑝̂ ~ Normal(𝜇𝑝̂ = .441 , 𝑆𝐸𝑝̂ = √
Consider all possible
random samples that could
have happened (sampling
distribution)
̂ (1−𝒑
̂)
𝒑
𝑛
)
.441 (1−.441 )
)
34
𝑝̂ ~ Normal(𝜇𝑝̂ = . 441, 𝑆𝐸𝑝̂ = . 𝟎𝟖𝟓)
Confidence Interval:
𝑝̂ ± 1.96 ∗ 𝑆𝐸𝑝̂
= .441 ± 1.96 ∗ (. 𝟎𝟖𝟓)
Question: Why do we use 1.96 in the formula?
= (. 𝟐𝟕𝟒, . 𝟔𝟎𝟖)
*Reichel and Morales, “Surveying immigrants without sampling frames – evaluating the success of alternative field methods,” Comparative
Migration Studies (2017) 5:1 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5209407/pdf/40878_2016_Article_44.pdf
**Huddleston, Thomas and Dag Tjaden, Jasper. ”Immigrant Citizens Survey: How immigrants experience integration in 15 European cities,” King
Baudouin Foundation and The Migration Policy Group. May 2012.
336
Critical Values
95% interval:
𝑝̂ ± 𝟏. 𝟗𝟔 ∗ 𝑆𝐸𝑝̂
𝑝̂ ± 𝟏. 𝟗𝟔 ∗ √
80% interval:
𝑝̂(1−𝑝̂)
𝑛
𝑝̂ ± (? ) ∗ 𝑆𝐸𝑝̂
𝑝̂ ± (? ) ∗ √
𝑝̂(1−𝑝̂)
𝑛
For a C% confidence interval (𝑝̂ ):
𝑧 ∗ : the __________________________ of the z-interval
• Tells you how many _________________________
you’re including in each side of your interval.
• Determines the _____________________.
𝑝̂ ± (𝑧 ∗ ) ∗ 𝑆𝐸𝑝̂
𝑝̂ ± (𝑧 ∗ ) ∗ √
𝑝̂ (1 − 𝑝̂ )
𝑛
How to get the critical value → use the standard normal curve (μ = 0, σ = 1)
The _________________ on the standard normal curve tells you the ____________________________
away from the center.
Find and interpret z* for 95% confidence level (show calculator input):
Find and interpret z* for 80% confidence level (show calculator input):
337
80% confidence interval (with mock data):
a) A random sample of 34 asylum-seekers had 15 adult men. Estimate the proportion of asylum-seekers
that are adult men, with 80% confidence.
b) Which is wider: the 95% or 80% interval? Why?
Confidence Interval Calculator Steps (one sample conf interval for a proportion)
STAT → TESTS →
A:1-PropZInt…
Input x, n, C-level
Calculate
338
Output
Sample Size Calculations
The mock data above had a low sample size (n = 34).
The sampling distribution and resulting confidence
interval is displayed to the left.
In the actual EU migration survey, researchers
randomly selected 583 Madrid immigrants and 257
were male
257
𝑝̂ =
= 44%
583
Same sample proportion (same point estimate),
higher sample size
Sample size intuition: a higher sample size (n) means…
• More data points per sample
• Unusual sample proportion values are _________________ (easy to get a ‘wonky’ sample when
you only collect a few data points)
• Less variability in our _______________________
• Confidence interval is ________________
Sample size calculations and the margin of error:
𝑝̂ ± margin of error
Standard error: measures variability of your
estimation process (specifically - the _________
______________ of sample proportions from
true proportion)
𝑧 ∗ ∗ 𝑆𝐸𝑝̂
Critical value – the number
of standard errors you
include in your interval
-Determined by the
__________________, not
the variability of your
estimation process
𝑧∗ ∗ √
𝑝̂(1−𝑝̂)
𝑛
Divide by n, so the higher n gets, the _________
_________________________ of your estimates
339
Example: You are designing the Immigrant Citizens Survey (data discussed earlier). You want to provide a
95% confidence interval for the proportion of recent immigrants who are adult men, and you anticipate
that 40% of immigrants will be adult men.
a) If you want your margin of error to be only 5%, how many recent immigrants should you randomly
sample (assume 40% of immigrants sampled are adult men)?
b) By what factor would you need to increase the sample size to cut the margin of error in half? You can
use your answer for part (a) as an example.
Discussion
Discussion Question: You are designing the Immigrant Citizens Survey (data discussed earlier). You want
to provide a 95% confidence interval for the proportion of recent immigrants who are adult men. You
have no prior data about this proportion. If you want to ensure your margin of error is no more than 1%,
how many recent immigrants should the study randomly sample?
340
How much of the Earth is covered by water?
What proportion of the Earth is covered by water? We will investigate this question by
taking a random sample of locations on the globe.
1. How many locations did your class sample? ______ How many locations were water? ______
2. Calculate the proportion of locations from your sample that are water. p̂ = _______
!
3. Construct a 95% confidence interval to estimate the proportion of the Earth that is water.
341
Estimating a Population Proportion:
Important ideas:
Check Your Understanding
A community activist group in Austin, Texas wanted a particular issue to be placed on the ballot of the
upcoming election. To make it on the ballot, 20,000 valid signatures were needed. The group turned in
their petition with 24,598 signatures. To pass the validity test 20,000/24,598 = 81.3% of the signatures
must be valid. It is too time consuming to check all of the signatures, so a random sample of signatures
are checked. The individual checking the signatures needs to be 95% confident that the true proportion
of valid signatures are estimated with, at most, a 2% margin of error.
1. Using a conservative estimate for 𝑝̂ , how large of a sample is needed?
2. In the activist group’s previous petition, 85% of the signatures were valid. Using this value as a
guess for 𝑝̂ , find the sample size needed for a margin of error of at most 2 percentage points
with 95% confidence. How does this compare with the required sample size from Question 1?
3. What if the company president demands 99% confidence instead of 95% confidence? Would
this require a smaller or larger sample size, assuming everything else remains the same?
Explain your answer.
342
Immigration
In October of 2018, a caravan of migrant asylum-seekers from central America traveled to the U.S.
There were thousands in the caravan, many escaping violence in their home countries. Liberalleaning news sources like MSNBC displayed mostly images of women and children in the caravan,
which evoked compassion and sympathy. Conservative-leaning news sources like Fox displayed
mostly images of single adult men acting aggressively, evoking fear and a need for security.
So, what proportion of asylum-seekers are single adult men vs. families? Unfortunately, the U.S.
doesn’t have very good data to analyze on this topic. But in Europe, good migration data has been
collected. We’ll use a sample from the population of EU migrants to create a one-sample zinterval for the proportion of EU migrants who are adult men.
The EU Migration Debate and Gender of Immigrants
Vital International
AP Photo/Michael
Svarnias
Depictions of migrants in the EU
range from sympathetic
portrayals of families escaping
violence to fear-inducing
portrayals of groups of single
adult men illegally crossing
borders.
“As you can see from this picture, most of the people coming are young
males and, yes, they may be coming from countries that are not in a very
happy state, they may be coming from places that are poorer than us, but the
EU has made a fundamental error that risks the security of everybody.”
– Nigel Farage, 2016 (U.K.)
“I said to myself, ‘Wow. They’re all men.’ You look at it. There are so few
women and there are so few children. And not only are they men, they’re
young men. And they’re strong as can be—they’re tough looking cookies. I
say, what’s going on here?”
– Trump, speaking about the EU while campaigning in 2015
Let’s test these claims by collecting our own sample!
The beads represent the immigrant population of Madrid, which has some of the best records* on
EU migration. The orange beads represent males. The blue beads represent non-males.
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
343
1. How many people did your class sample? ______ How many people were male? ______
2. Calculate the proportion of people from your sample that were male. p̂ = _______
!
3. Construct a 95% confidence interval to estimate the proportion of immigrants who are male.
4. The Immigrant Citizens Survey** conducted a random sample of recent migrants in Madrid’s
comprehensive database. 257 of the 583 immigrants sampled were male. Without performing the
whole procedure, describe how you believe their confidence interval would differ from the interval
above.
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
344
AP Statistics Handout: Lesson 8.3
Topics: one vs. two sample inference, two-sample z-interval for a difference of proportions
Guided Notes
Context for Today’s Lesson: The Race/Resumé Study
In this study*, investigators created mock identical resumés, which were sent to job placement ads in
Chicago and Boston. Each resumé was randomly assigned either a commonly-white or commonly-black
name. In total, 246 out of 2445 commonly-white named resumés received a callback and 164 out of
2445 commonly-black named resumés received a callback.
*Bertrand, Marianne and Sendhil Mullainathan. "Are Emily And Greg More Employable Than Lakisha
And Jamal? A Field Experiment On Labor Market Discrimination," American Economic Review, 2004,
v94(4,Sep), 991-1013. https://www.nber.org/papers/w9873
One vs. Two Sample Inference
Study results
CommonlyWhite Names
CommonlyBlack Names
Total
Called back
246
164
410
Not called back
2199
2281
4480
Total
2445
2445
4890
Find the following quantities (show
calculations): 𝑛1 , 𝑛2 , 𝑝̂1, 𝑝̂ 2 , 𝑝̂1 − 𝑝̂ 2
̂𝟏 > 𝒑
̂𝟐
If there’s hiring discrimination, 𝒑
Group 1: White
𝑝̂1 = proportion of commonly-white
name apps that got callback.
𝑝̂1 = _________
>
Group 2: Black
𝑝̂2 = proportion of commonly-white
name apps that got callback.
𝑝̂2 = _________
Are these proportions _____________________________ to show discrimination, or could this
difference have been a result of _________________________?
345
One vs. Two Samples
One-sample situations: you compare a statistic in __________ population against a ____________about
that population.
-Ex: Is the proportion of recent EU migrants who are male
actually lower than the claimed 75%?
Two-sample situations: you measure the ________________________ in _____________________
_____________________ and see if they are significantly different.
-Ex: Is the proportion of callbacks for commonly-white
name apps higher than for commonly-black name apps?
Two-Sample Z-Interval for a Difference of Proportions
Hypotheses: We don’t need to set up hypotheses to construct a confidence interval, but doing so is going
to help us conceptualize why our interval is useful.
𝐻0 : 𝑝1 = 𝑝2
𝐻𝐴 : 𝑝1 > 𝑝2
The null (______________) hypothesis: there is ___________________,
so the callback rate is the same in both groups. You’re seeing if there’s
evidence to reject this claim.
The alternative (________________) hypothesis: there is discrimination,
in which case the commonly-white named applications received a
________________________ of callbacks.
Rewrite these hypotheses in a more mathematically convenient way:
Where:
𝑝1 is the proportion of _______ applicants with commonly-white names who’d receive callbacks
when applying to jobs like the ones in this study.
𝑝2 is the proportion of _______ applicants with commonly-black names who’d receive callbacks
when applying to jobs like the ones in this study.
Making the interval
Under certain conditions:
𝑝̂2 = _________
̂1 )
𝑝̂ (1−𝑝̂ )
~ Norm(𝜇 = 𝑝̂1 - 𝑝̂2 , 𝜎 = √𝑝̂1 (1−𝑝
+ 2 𝑛 2)
𝑛
1
2
(1−.101)
.067 (1−.067)
~ Norm(𝜇 = .034, 𝜎 = √.1012445
+
)
2445
~ Norm(𝜇 = .034, 𝜎 = 0.0079)
Confidence Interval: .034 ± 1.96(0.0079)
(1.85%, 4.95%)
346
Confidence Interval: (1.85%, 4.95%)
Interpretation: We are __________________ the interval from _____________________ captures the
________________________ in proportion of callbacks for resumés with commonly white vs. black
names (among jobs similar to the ones in this study).
Significance: Since ________________ our interval, it’s _____________________ to assume 𝑝1 = 𝑝2 .
Therefore, we have convincing evidence that commonly white name resumés receive a _____________
callback rate (among jobs similar to the ones in this study).
Formula for confidence interval:
95% Interval
95% Confidence
Interval Formula
C% Interval
statistic ± margin of error
C% Confidence
Interval Formula
𝑝̂ (1 − 𝑝̂ )
𝑛
95% One-Sample 𝑝̂ ± 𝟏. 𝟗𝟔 ∗ √
Interval Formula
C% One-Sample
Interval Formula
𝑝̂1 (1 − 𝑝̂1 ) 𝑝̂2 (1 − 𝑝̂2 )
95% Two-Sample
(𝑝̂1 - 𝑝̂2 ) ± 𝟏. 𝟗𝟔 ∗ √
+
Interval Formula
𝑛1
𝑛2
C% Two-Sample
Interval Formula
𝟏. 𝟗𝟔: the critical value for a 95% z-interval. It means
we’re include ~2 standard errors in the interval
statistic ± margin of error
𝑝̂ (1 − 𝑝̂ )
𝑝̂ ± 𝒛∗ ∗ √
𝑛
(𝑝̂1 - 𝑝̂2 ) ± 𝒛∗ ∗ √
𝑝̂1 (1 − 𝑝̂1 ) 𝑝̂ 2 (1 − 𝑝̂2 )
+
𝑛1
𝑛2
𝒛∗: the critical value of the z-interval. Tells you how
many standard errors you’re including in your interval.
In the Bertrand-Mullainathan race/resumé study, mock identical resumés were sent to job placement ads
in Chicago and Boston. Each resumé was randomly assigned either a commonly-white or commonlyblack name. In total, 246 out of 2445 commonly-white named resumés received a callback and 164 out
of 2445 commonly-black named resumés received a callback.
a) Construct and interpret a 95% confidence interval for the difference in callback rates for commonly
white vs. black named resumes (among jobs like the ones in this study).
b) Use your interval to draw a conclusion about whether callback rates differ by commonly white vs.
black name.
347
Lesson 8.3 Discussion
Statistically Significant vs. Practically Important
Statistically Significant Difference: Difference is too large to arise by ____________________.
• Confidence intervals and hypothesis tests tell us this
Practically Important Difference: Difference is large enough to have _____________________________
in the world.
• Descriptive statistics and our experiences tell us this.
Let’s look at the study results one more time:
CommonlyWhite Names
CommonlyBlack Names
Total
Called back
246
164
410
Not called back
2199
2281
4480
Total
2445
2445
4890
𝑝̂1 = 10.1%
𝑝̂2 = 6.7%
𝑝̂1 − 𝑝̂2 = 3.4%
Confidence interval for difference in
callback rates: (1.8%, 4.9%)
Discussion Question: We found the difference between callback rates was statistically significant, but is
it practically important? Look at the data, the callback rates, and the final confidence interval. Do you
think the margin between black/white callback rates is practically important? Explain your reasoning.
348
8.3 PRACTICE!
Presidential Approval
Many news organizations conduct polls asking adults in the United States if they approve of the job the
president is doing. How did President Obama’s approval rating change from August 2009 to September 2010?
According to a CNN poll of 1024 randomly selected U.S. adults on September 1-2, 2010, 50% approved of
Obama’s job performance. A CNN poll of 1010 randomly selected U.S. adults on August 28-30, 2009, showed
that 53% approved of Obama’s job performance. Use the results of these polls to construct and interpret a 90%
confidence interval for the change in Obama’s approval rating among all US adults. Based on your interval, is
there convincing evidence that Obama’s job approval rating has changed?
349
Which grade is more likely to go to Prom?
At many high schools, Prom is an annual dance that only Juniors and Seniors can purchase
tickets for. The student council at a large high school is wondering if Juniors or Seniors are
more likely to attend Prom. They take a random sample of 50 Juniors and find that 28 are
planning on attending Prom. They select a random sample of 45 Seniors and 29 are
planning on attending. Construct and interpret a 95% confidence interval for the difference
in proportions of Juniors and Seniors who are planning on attending Prom.
What is the point estimate for…
the proportion of Juniors planning on attending prom? 𝑝
"! = ________
"" = ________
the proportion of Seniors planning on attending prom? 𝑝
the difference in the proportion of Jrs and Srs planning on attending prom? 𝑝
"! − 𝑝
"" = ________
1. Construct and interpret a 95% confidence interval for the difference in proportions of Juniors
and Seniors who are planning on attending prom.
2.
Does the interval provide convincing evidence that Juniors have a lower proportion planning
on going to prom or is it plausible that there is no difference between the two classes?
Explain.
350
Constructing a Confidence Interval for p1 – p2
Important ideas:
Check Your Understanding
In a social study, a random sample of 150 teachers were selected and an independent random sample
of 100 nurses were selected. Each person was asked if they currently have a second job. The results
showed that 48 of the 150 teachers and 21 of the 100 nurses had a second job. Construct and interpret
a 95% confidence interval for the difference in the proportion of all teachers and nurses that have a
second job.
351
AP Stats Chapter 8 Formula Study Sheet
Lesson
One Sample Proportion
Symbol for statistic
Symbol for parameter
Name of the procedure
RANDOM condition
Independent condition
NORMAL condition
Formula for standard error
Formula for margin of error
General formula for confidence
interval
Specific formula for confidence
interval
352
Two Sample Proportion
Sample size practice
1. Edgar Martinez and Ingrid Gustafson are the candidates for mayor in a
large city. We want to estimate the proportion p of all registered voters in
the city who plan to vote for Gustafson with 95% confidence and a margin
of error no greater than 0.03. How large a random sample do we need?
2. High school students who take the SAT Math exam a second time
generally score higher than on their first try. Past data suggest that the
score increase has a standard deviation of about 50 points. How large a
sample of high school students would be needed to estimate the mean
change in SAT score to within 2 points with 95% confidence?
3. A college student organization wants to start a nightclub for students
under the age of 21. To assess support for the idea, the organization will
select an SRS of students and ask each if he or she would patronize this
type of establishment. What sample size is required to obtain a 90%
confidence interval with a margin of error of at most 0.04 if they suspect
pˆ = .75?
353
1. Suppose we are interested in finding out the proportion of the population at EKHS that
has seen The Office. We contact an SRS of 100 students in the school. Of these 100
students, 63 report seeing The Office. Find a 95% confidence interval for the true
proportion of EKHS students who have seen The Office.
2. Mrs. Statistics was an all-star basketball player in high school. To prove that she still has
skills, she took 50 free throws and made 31 of them. Think of these 50 shots as being a
random sample of all the free throws she has ever taken. Find a 99% confidence interval for
the true proportion of free throws Mrs. Statistics would make.
354
355
2010 AP® STATISTICS FREE-RESPONSE QUESTIONS
3. A humane society wanted to estimate with 95 percent confidence the proportion ofhouseholds in its county that
own at least one dog.
(a) Interpret the 95 percent confidence level in this context.
The humane society selected a random sample of households in its county and used the sample to estimate the
proportion of all households that own at least one dog. The conditions for calculating a 95 percent confidence
interval for the proportion of households in this county that own at least one dog were checked and verified, and
the resulting confidence interval was 0.417 ± 0.119.
(b) A national pet products association claimed that 39 percent ofall American households owned at least one
dog. Does the humane society's interval estimate provide evidence that the proportion ofdog owners in its
county is different from the claimed national proportion? Explain.
(c) How many households were selected in the humane society's sample? Show how you obtained your answer.
356
Unit 6
Chapter 9
Testing Claims
About Proportions
357
AP Statistics Handout: Lesson 9.1
Topics: hypotheses, logic of hypothesis tests, p-values and alpha levels
Guided Notes
Flint Water Crisis:
Samuel Wilson
(MLive.com)
Jake May (The
Flint Journal,
AP Images)
WNEM Newsroom
2014
2014/2015
2015
Flint city officials celebrate as they
cut costs by switching the water
supply from Detroit to the Flint River.
Residents voice concerns
about declining water quality.
Mayor drinks tap water on local
television to “prove” the water is safe.
Outside investigators start testing
Flint’s water systems.
Hypotheses
Null hypothesis (𝐻0 ): The __________________ belief about a parameter’s value.
• Often called the “________” hypothesis – nothing new or interesting happening
• Assumed ___________________________, until evidence convinces us otherwise
Alternative hypothesis (𝐻𝐴 ): Our unproven belief about a _______________________________, for
which we have to gather evidence.
• Often called the “___________________” hypothesis – it’s what our studies aim to demonstrate
• Not assumed to be true: we have to ____________________________________ to support it.
Court System Analogy
Person on trial is assumed to be innocent (the default/dull belief - 𝐻0 ), until proven to be guilty through
evidence (𝐻𝐴 )
Two possible outcomes:
• There’s enough evidence to reject their innocence (Reject 𝐻0 )
• There’s not enough evidence to reject their innocence (Fail to Reject 𝐻0 )
Note: you ________________________. It’s assumed to be true already, so our research isn’t proving it.
• This is why you’re declared “_______________________” rather than “innocent” in court →
your innocence was assumed at the beginning.
358
Flint Hypotheses
Non-numerical version:
𝐻0 : Flint’s water is _____________ to drink
• Default/dull belief (___________________)
𝐻𝐴 : Flint’s water is ________ safe to drink
• Unproven research hypothesis we aim to
demonstrate (_______________________)
Numerical version:
𝐻0 :
𝐻𝐴 :
p: true proportion of all homes in Flint
with high lead content in their water
According to EPA regulations, if ___________________________ of homes in a city have high lead
content in their water (>15 parts per billion), the city’s water supply is unsafe.
Logic of Hypothesis Tests
Hypothesis tests are statistical jiu-jitsu:
• You assume the _________________
• You calculate how
___________________ your sample
data is if 𝐻0 is true
• If it’s unlikely enough, you
_______________
You assume 𝐻0 to be true in order to
eventually ___________________________.
Flint hypothesis test logic (numeric example):
Goal: Find enough statistical evidence to _______________ and
support the ________
𝐻0 : p = 0.10
𝐻𝐴 : p > 0.10
1. Start by assuming the null (the default) ____________.
• We assume that water in Flint is safe by EPA standards:
_________________________ of all homes have lead levels
above 15 ppb.
p: true proportion of all homes in
Flint with high lead content in their
water
2. Ask yourself: if the null is in fact true, how ___________ is the data that you’ve gathered?
• I randomly sampled water from 100 homes and 30 of them had high lead content (30%). This sample
______________________ under my earlier assumption that only 10% of homes in the whole city
had high lead content.
3. Draw a conclusion:
• Under my assumption that only 10% of homes have high lead content, the actually observed data
(30% of 100 homes had high lead content) is highly unlikely. So, ____________________________
___________________________. There’s convincing evidence that __________________________
of homes have high lead content.
359
p-values and Alpha (α) Levels
What is “unlikely?”
𝐻0 : p = 0.10
𝐻𝐴 : p > 0.10
For each case, comment on the conclusion we’d draw about our hypotheses:
Case A: Sample 1000 homes, 300 have high lead. 𝑝̂ =
Case B: Sample 10 homes, 2 have high lead. 𝑝̂ =
2
10
300
1000
= 0.30
= 0.20
Case C (real study* data): Sample 252 homes, 42 has high lead. 𝑝̂ =
42
252
= 0.17
p-value: The ____________________ of observing your measured statistic value (or one more extreme)
if you ____________________________________.
o Stands for “probability value”
o Helps us determine how ___________________ our sample data is under the null
Is there convincing statistical evidence that the Flint water system is unsafe? A Virginia Tech study*
42
randomly sampled water from 252 homes in Flint. Of those, 42 had high lead content. 𝑝̂ =
= 0.17
252
This phrase means you should do
a _____________________
𝐻0 : p = 0.10
𝐻𝐴 : p > 0.10
If it’s true that only 10% of all homes have high lead content (𝐻0 ), the probability that 17% (or more) of
the sampled homes would have high lead content is only ________
Since the p-value is _______________, we have serious doubts about our null
hypothesis assumption. We can reject the null hypothesis!
*Edwards, M., et al. Virginia Tech’s Flint Water Study (published Sept. 2015, accessed July 2020):
http://flintwaterstudy.org/information-for-flint-residents/results-for-citizen-testing-for-lead-300-kits/
360
_______________
p-values, alpha levels, and conclusions
When the p-value is low…
• The data we actually observed is ________________ under the null hypothesis assumption
• We can _______________________ assumption
When the p-value isn’t that low…
• The data we actually observed ______________________ under the null hypothesis assumption
• We ________________________ the null assumption
Significance level (α): the ‘___________________’ for p-values at which we reject the null hypothesis
• Commonly, we use α = 0.05
Rules:
(p-value < α) → we reject 𝐻0 → _________________________________ of 𝐻𝐴
(p-value ≥ α) → we fail to reject 𝐻0 → ______________________ convincing evidence of 𝐻𝐴
Conclusions template: Because our p-value (____) is less/greater than our alpha level (___), we
reject/fail to reject 𝐻0 . We do/don’t have convincing evidence that (𝐻𝐴 in context).
Write the full conclusion: Is there convincing statistical evidence that the Flint water system is unsafe?
361
Lesson 9.1 Discussion
You: The p-value is low, so we have convincing evidence the lead levels are unsafe!
The Mayor: What are you talking about? I tested my own home’s water, and it’s fine.
You: But statistically…
The Mayor: Statistically, I know every person on my block still drinks their water. It’s
perfectly fine!
WNEM Newsroom
Discussion Question: How would you explain what this p-value means to someone who doesn’t know
any statistics? Be clear and specific.
362
Is Mrs. Gallas a good free throw shooter?
Mrs. Gallas claims she is an 80% free throw shooter. To prove her skills she shoots 50 free
throws and makes 32 shots. Is Mrs. Gallas exaggerating about her free throw skills?
1. Identify the population, parameter, sample and statistic.
Population:
Parameter:
Sample:
Statistic:
2. There are two possible explanations for why Mrs. Gallas only made 32/50 shots.
1.)
2.)
To test Mrs. Gallas’ claim, we will assume #1, she is an 80% free throw shooter, and
examine the likelihood that she makes 32/50 shots through simulation.
3. Use the spinner provided to simulate 50 free throws shot by an 80% free throw shooter
by spinning 50 times. What is your sample proportion of shots made?
4. Repeat for another sample of 50 spins. Calculate the sample proportion.
5. Add your sample proportions to the dotplot on the board. Each person in your group
should add two dots to the board. Sketch the dotplot below.
proportion of FT made
363
6. What does each dot represent?
7. One student says, “Each dot represents the proportion of free throws made out of 50 free
throws shot by Mrs. Gallas.” Is this correct? Explain.
8. What percentage of the dots represent a percentage of 64% or less?
Interpret this percentage in context.
9. Based on your answer to Question 8, does the observed p̂ = 0.64 result give convincing
evidence that Mrs. Gallas is exaggerating? Or is it plausible that an 80% shooter can have a
performance this poor by chance alone?
364
AP Statistics Handout: Lesson 9.1 part 2
Topics: bias vs. error, Type I/II errors, problem of multiple tests, power
Guided Notes
Poll after poll after poll predicted Hillary Clinton to trounce Donald Trump in the 2016
election. Yet…
Actual election results from Nov 8th, 2016:
Statistical predictions from Nov 7th, 2016
(chance of winning):
Trump wins electoral college
One of the biggest surprises: Wisconsin
Every 2016 Wisconsin poll shows lead for Clinton
Voted for democrat every election since 1992
Election Day 2016:
Trump Wins Wisconsin
365
Bias vs. Error
Bias: when our estimators _____________________________ under or overestimate a parameter value.
Error: when an _______________ estimator produces an estimate that slightly “misses its mark” due to
____________________________ in the sampling process.
Example: Among Wisconsin voters planning to vote for a major party candidate (Clinton or Trump),
what proportion will vote for Trump? Imagine 60% of all Wisconsin major party voters will vote for
Trump.
Welcome to Theory Land – “The not real place.”
All Wisc. major party voters
Realm of Perfect Parameters
Realm of Sucky Statistics (n = 30)
What proportion of all Wisconsin major party voters
will vote Trump? Let’s get a sample!
Random sample possibility #1:
Random sample possibility #2
Random sample possibility #3:
Resulting sampling distribution (all possible random samples of size n = 30):
Unbiased sampling method: doesn’t tend to
systematically under/overestimate
(_____________________ at true parameter value).
A random sample (from an unbiased estimation process) produces an underestimate (an error):
By chance, we randomly sampled fewer Trump voters
than expected. This produces ________________.
This is why we give confidence intervals ___________
____________________. For unbiased estimators, we
still expect errors because of chance variation in our
random sampling process.
366
Were the Wisconsin polls systematically biased or did they underestimate Trump by chance alone (i.e.
make an error)?
If the polls underestimated Trump by chance alone…
Polls would ___________ at true proportion of Trump voters, with
some errors __________________ due to chance variation in
sampling. Most would capture true proportion
___________________________.
What we actually saw from Wisconsin polls…
Why were Wisconsin polls biased? Several sources of possible bias…
1. Nonresponse bias: Education
• Less educated people are less responsive to polls.
• Usually this isn’t a big problem since less educated people tend to split their votes, but this time
they voted heavily towards Trump. The polls didn’t adjust for this and underestimated Trump’s
vote totals.
• Effect was big in Wisconsin, since state has many white voters without college degrees.
2. Undercoverage bias: Undecided Voters
• Polls cannot figure out how undecided voters will cast their ballots on election day.
• Usually, this is not a problem since undecided voters tend to split votes between candidates.
• Some results suggest that undecided voters broke more towards Trump than expected in
Wisconsin, so these votes were undercovered by the polls.
3. Other potential bias from higher than expected Trump voter turnout and lying to pollsters (although
these theories have less evidence than the first two).
Why do we always have to check conditions? → One reason: making estimates (confidence intervals) or
evaluating claims (hypothesis tests) based on biased sampling is __________________.
• Our inference methods account for _____________. They don’t account for ___________.
• The random condition is one way to check for bias.
367
Type I and Type II Errors
Let’s take an unbiased situation: It’s November 8th, 2016. Voting just closed in Wisconsin and you want
to quickly estimate the proportion of voters who voted for Clinton. It would take a long time to count all
the votes, so you get an SRS of 500 votes from all Wisconsin ballot boxes.
According to many election experts, candidates that get 47% or more of the vote tend to win elections
(about 3% of votes go to third party candidates or “Mickey Mouse”). Of the 500 sampled votes, 255
were votes for Clinton. Use a hypothesis test to determine if there is convincing evidence that Clinton
will win Wisconsin
Hypotheses:
𝐻0 : ____________
𝐻𝐴 : ____________
Where p is the __________________ of voters who will choose Clinton.
Significance Level: α = 0.05
Calculations:
Let’s assume 𝐻0 is true: exactly 47% of Wisconsin will vote Clinton. We get the following sampling
distribution for random samples of 500 voters each (n = 500):
α = 0.05 → will only reject with a p-value of 0.05
or less
5% chance of getting an unusual sample that
makes us reject 𝑝0 _______________________!
Should we actually reject 𝒑𝟎 and conclude
_____________________ Wisconsin? Or did we
just get an unusually Clinton-supportive sample
by ________________________?
Type I Error
If the Clinton voter rate is truly 0.47, the probability of getting a sample proportion of .51 is less than 5%.
So, we reject 𝐻0 and conclude Clinton will win Wisconsin.
But she ___________________ Wisconsin!
By chance, the 500 people we randomly sampled happened to be more supportive of Clinton than the
__________________________. We falsely rejected 𝐻0 because of __________________________.
Type I Error: When you _____________________ but 𝐻0 is actually true.
• Type I Errors happen α% of the time (so, for most tests, 5% of the time we make a Type I error)
• α (the significance level) is the ______________________________.
• If you are scared of a Type I error, ________________________ to below 0.05. This makes
rejecting the 𝐻0 more difficult, but it also ______________________________.
368
Type II Error
Type II Error: When you fail to reject 𝐻0 but 𝐻0 _________________________. In other words, you
should have rejected it but didn’t.
Problem of Multiple Tests
Scenario:
• Imagine you really want Clinton to win Wisconsin. You take one random sample of 500 ballots
230
and find 230 voted for Trump (𝑝̂ =
= 46%).
500
•
You know Wisconsin usually votes Blue, so you think this must be an unusual sample. You take
215
another sample of size 500. In this one, 215 voted for Clinton (𝑝̂ =
= 43%).
•
You keep taking samples until finally you find one where 257 out of 500 voted for Clinton (𝑝̂ =
257
= 51.4%). You conclude there is statistically significant evidence Clinton will win!
500
500
What is the problem with this scenario?
The Problem of Multiple Tests
• When you run a test and sample many times, you may be “p-hacking” → fishing for a sample
that will prove your point, even though all you’re really doing is
___________________________________.
• If 𝛼 = 0.05, then 5% of the time we’ll get a falsely significant result from an unusual sample
(_________________________).
1
• 0.05 = → this means that roughly __________________ we take, we’ll get a significant result
20
by chance. So if you take samples over-and-over again, you’re bound to get a significant result
eventually, ____________________________.
• This is called the ___________________________________.
Avoiding the Problem of Multiple Tests
Get one sample of the highest sample size you can muster → whatever results from that sample are
your final results.
______________________________ or do many ________________ tests. If you have to repeat a test,
____________________ level.
369
Power
Power: the __________________ of correctly rejecting 𝐻0 when it’s false.
Effect size: the __________________ between a proposed null hypothesis value and the actual
parameter value.
Power is the __________________ of the Type II Error rate. It’s the probability of correctly rejecting 𝐻0
when it’s false (Type II Error is failing to reject 𝐻0 when it’s false).
Type II Error Rate = _________________
Usually, you want a test to have ______________________: so you know that if 𝐻0 is not true, you have
a __________________________ of rejecting it.
Example: New Drug to Lower Blood Pressure
Your research team is testing the effectiveness of a new systolic blood pressure drug—Drug B. The
doctors on the research team say that if you can lower systolic blood pressure by 10 points or more,
such a difference will have meaningful health benefits. If you lower blood pressure by fewer than 10
points, it’s not a medically meaningful difference.
Blood pressure changes are distributed normally and have a standard deviation of about 21.5 points
between people. You give the drug to a group of 20 subjects. What is your power to detect a true
average blood pressure drop by 10 points?
Hypotheses: Where μ = average difference in blood pressure after taking the drug.
𝐻0 : ___________
𝐻𝐴 : ___________
Significance Level: α = 0.05
Calculations: Start by assuming 𝑯𝟎 is true and determine the rejection zone
Any sample mean with a blood pressure drop
greater in magnitude than 7.88 points should
result in _________________________ the null.
α = 0.05
Next - Assume 𝑯𝑨 is true and see what sample means we’re likely to get…
The ____________ of this test to detect a 10
point drop in blood pressure is 67%--if the true
__________________ is -10, we will reject 𝐻0
only 64% of the time.
370
Original power: 67%
α = 0.05
Increase power method 1: increase α
α = 0.15
Problem: Recall that α is the _________________
________. So even though increasing α increases
the power, it also increases the ___________ of a
Type I error. So, don’t increase α just to increase
power!
Increase power method 2: raise effect size
True average blood pressure drop from Drug B is
now 13 points!
Problem: _________________ to raise drug’s
effect size without changing the drug itself!
Increase power method 3: raise sample size (n)
Let’s raise the sample size (n) so that our estimates
become more _______________ (i.e. less chance
of an unusual sample).
Usually raising n is the _________ way to increase
power.
Why it makes sense that raising the sample size would also raise the power:
-If a drug is effective with 3 people, you may ___________________________ it will work for everyone.
-If a drug is effective with 10,000 people, you will be more sure it’s ____________________ effective.
-If the drug is effective, increasing the sample sizes makes you ____________________ to detect its
effectiveness because your estimates are more precise.
Summary: How to increase power
1. Increase α level  _______________ (most of the time)
2. Increase effect size  __________________ (most of the time)
3. Increase sample size (n)  ___________ (most of the time)
371
Lesson 7.6 Discussion
Comic from XKCD: xkcd.com/882
Discussion Question: What statistical problem is this cartoon pointing out? How might this problem be
an issue for scientists testing new ideas, medicines, or theories?
372
Is this gender discrimination?
A local engineering firm had to conduct a series of layoffs recently. The company has 180
employees that could be laid off and they will choose 10. All are equally qualified so the
company decides to use a lottery system to be carried out by the manager to decide who
will be laid off. The manager posts a list of the employees to be laid off. Five employees are
women and five are men. One of the women claims this is gender discrimination and starts
a lawsuit against the company.
1. The manager responds, “How could there be gender discrimination when half of the
employees laid off were female and half were male?” What additional information do
you need to evaluate this statement?
2. How can you use a simulation to investigate the gender discrimination claim? Detail a
process that could be used.
3. Perform one repetition of the simulation. Proportion of females in sample = _______
4. Add your proportion to the class dotplot. Copy below.
5. What percentage of the dots represent half or more females being laid off?
6. Interpret this percentage in context.
7. Do you have convincing evidence of gender discrimination? Explain.
373
Significance Tests: The Basics
Important ideas:
Check Your Understanding
Factinate.com claims that 84% of teenagers think highly of their mother. To investigate this claim, a
school psychologist selects a random sample of 150 teenagers and finds that 135 think highly of
their mother. Do these data provide convincing evidence that the true proportion of teens who think
highly of their mother is greater than 0.84?
1. State appropriate hypotheses for performing a significance test. Be sure to define the parameter
of interest.
The school psychologist performed the significance test and obtained a P-value of 0.0225.
2. Explain what it would mean for the null hypothesis to be true in this setting.
3. Interpret the P-value.
4. What conclusion would you make at the α = 0.05 level?
374
Are you sure Mrs. Magic isn’t a good free throw shooter?
In our introduction to significance tests, we used simulation to estimate a P-value to
decide whether or not Mrs. Magic as exaggerating about her free throw percentage.
Today, we will use a formula to find a P-value.
1. We’re going to carry out the significance test from lesson 9.1 again. Begin by writing
the hypotheses.
2. a. Each class found a different P-value because each dotplot was different. Would it be
appropriate to use a Normal distribution to model the sampling distribution of p̂ ?
Justify your answer.
b. Are there any other conditions we should check?
3. Now that conditions have been met, find the mean and standard deviation of the
sampling distribution of p̂ .
4. Use the mean and standard deviation you found to label the Normal curve.
5. How many standard deviations below the
mean (z-score) is pˆ = 0.64 ? Label it on
the normal curve.
6. Find the probability of an 80% shooter
making 32/50 ( pˆ = 0.64 ) or less.
7. What conclusion can we make?
375
Significance Test for p
Important ideas:
Check Your Understanding
Sharon claims that 90% of students can identify the smell of a skunk. She carries out a study to test
this theory. She selects a random sample of 100 students and asks them each to take a whiff from a
bag that is filled with skunk smell. She finds that 84 are able to correctly identify the smell as that of
a skunk. She would like to know if these data provide convincing evidence that less than 90% of
students can identify the smell of a skunk. Use α = 0.05.
a. State appropriate hypotheses for performing a significance test. Be sure to define the
parameter of interest.
b. Explain why the sample result gives some evidence for the alternative hypothesis.
c. Check if the conditions for performing the significance test are met.
d. Calculate the standardized test statistic and P-value.
e. What conclusion should Sharon make?
376
AP Statistics Handout: Lesson 9.2
Topics: one-sample z-test for a proportion, four-step procedure, two-sided tests
Guided Notes
Flint Water Crisis:
Samuel Wilson
(MLive.com)
Jake May (The
Flint Journal,
AP Images)
WNEM Newsroom
2014
2014/2015
2015
Flint city officials celebrate as they
cut costs by switching the water
supply from Detroit to the Flint River.
Residents voice concerns
about declining water quality.
Mayor drinks tap water on local
television to “prove” the water is safe.
Outside investigators start testing
Flint’s water systems.
One-sample z-test for a proportion
Researchers’ claim: the new water from the Flint river corrodes pipes, so lead gets into the water.
• Among many other side effects, ingesting lead can hurt children’s brain development
According to EPA regulations, if ___________________________ of homes in a city have high lead
content in their water (>15 parts per billion), the city’s water supply is unsafe.
Is there convincing statistical evidence that the Flint water system is unsafe? A Virginia Tech study*
42
randomly sampled water from 252 homes in Flint. Of those, 42 had high lead content. 𝑝̂ =
= 0.17
252
Hypotheses
Non-numerical version:
𝐻0 : Flint’s water is _____________ to drink
• Default/dull belief (___________________)
𝐻𝐴 : Flint’s water is ________ safe to drink
• Unproven research hypothesis we aim to
demonstrate (_______________________)
Numerical version:
𝐻0 :
This phrase means you should
do a _____________________
𝐻𝐴 :
p: true proportion of all homes in Flint
with high lead content in their water
*Edwards, M., et al. Virginia Tech’s Flint Water Study (published Sept. 2015, accessed July 2020):
http://flintwaterstudy.org/information-for-flint-residents/results-for-citizen-testing-for-lead-300-kits/
377
Under the hood: calculating z-test statistics and p-values for a one proportion test
Goal: Find enough statistical evidence to reject the 𝑯𝟎 and
support the 𝑯𝑨
𝐻0 : p = 0.10
𝐻𝐴 : p > 0.10
1. Start by assuming the null (the default) is true.
• We assume that water in Flint is safe by EPA standards:
no more than 10% of all homes have lead levels above
15 ppb.
p: true proportion of all homes in
Flint with high lead content in their
water
1. We assume _______________________
in the population of flint have high lead
water.
p = 0.10
2. Ask yourself: if the null is in fact true, how likely is the
data that you’ve gathered?
2. The flint study collected water from 252
homes (n = 252). At this sample size, what
estimates for p are we _______________?
Let’s find the ________________________ of
𝑝̂ ’s from all possible random samples.
Under certain conditions:
𝑝̂ ~ Normal(𝜇𝑝̂ = 𝑝0 , 𝜎𝑝̂ = √
𝑝0 (1−𝑝0 )
𝑛
)
𝑝̂ ~ Normal(𝜇𝑝̂ = _____, 𝜎𝑝̂ = _______)
Now we look at the data we gathered (sample).
̂ = _________
42 out of 252 homes had high-lead water: 𝒑
Under the null assumption, how
̂)?
unlikely was the data we observed (𝒑
Two ways to measure that:
1. z-test statistic
2. p-value
378
Using the z-test statistic:
Z-Scores
Shows how many σ a data value
is above/below the mean.
Z-Test Statistics
Shows how many σ a ___________ (sample) is above/below _____.
𝑧=
data value − mean
𝑧=
standard deviation
z=
𝑥𝑖 − µ
z=
𝜎
𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑡 − 𝑛𝑢𝑙𝑙 𝑣𝑎𝑙𝑢𝑒
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑝̂−𝑝0
𝜎𝑝
̂
→ z=
𝑝̂−𝑝0
𝑝 (1−𝑝0 )
√ 0
𝑛
The ______________ the test statistic (in magnitude), the more _____________ your measurements are
under the null assumption. You start to ________________________________.
𝑧=
𝑝̂ − 𝑝0
=
𝜎𝑝̂
𝑧=
If 2 standard deviations away from 𝑝0
would be unlikely, then 3.5 standard
deviations away is ___________________.
Using the p-value: exactly how unlikely was the proportion we sampled?
a) This probability is the p-value. Explain why:
b) Show the calculator commands and final
result for the p-value:
3. Draw a conclusion:
Under my assumption that only 10% of homes have high lead content, the actually observed data
(16.7% of 252 homes had high lead content) is highly unlikely (__________________). So, ____________
_________________________. There’s convincing evidence that ___________________of homes have
high lead content.
379
According to the EPA, no more than 10% of homes in a water system should have high lead levels (>15
parts per billion). A Virginia Tech study* randomly sampled water from 252 homes in Flint. Of those, 42
42
had high lead content. 𝑝̂ =
= 0.167
252
Is there convincing statistical evidence that the Flint water system is unsafe?
Calculator steps for one-sample z-test for a proportion:
𝑝0 : null value
x: # of ‘successes’
n: sample size
Prop: 𝐻𝐴 direction
STAT → TESTS →
5: 1-PropZTest…
Calculate
380
Output
Two-Sided Tests
One-Sided Test
Two-Sided Test
According to the EPA, no more than 10% of
homes in a water system should have high lead
levels (>15 parts per billion). A Virginia Tech
study* randomly sampled water from 252 homes
in Flint. Of those, 42 had high lead content.
42
𝑝̂ =
= 0.167
252
Is there convincing statistical evidence that the
Flint water system is unsafe?
A city official claims that 10% of all homes in Flint
have high lead content in their water. A Virginia
Tech study* randomly sampled water from 252
homes in Flint. Of those, 42 had high lead content.
42
𝑝̂ =
= 0.167
252
Is there convincing statistical evidence that the
percent of homes with high lead content differs
from the official’s claim?
“Differs” means that the percent of
homes with high lead content is
_______________________ than 10%.
“Unsafe” means that _________ of
homes have high lead content in water.
Conceptual Hypotheses
𝐻0 : Flint’s water is ___________to drink
𝐻𝐴 : Flint’s water is _______ safe to drink
Formal Hypotheses
𝐻0 : p = 0.10
𝐻𝐴 : p ___ 0.10
Conceptual Hypotheses
𝐻0 : The proportion of homes in Flint with high lead
content ____ 10%
𝐻𝐴 : The proportion of homes in Flint with high lead
content _________ 10%
Testing whether __________
_____ of all homes have high
lead content in their water.
Formal Hypotheses
𝐻0 : p = 0.10
𝐻𝐴 : p ___ 0.10
One-sided test: when you’re testing an alternative
in ____________________ (𝐻𝐴 either uses < or >)
Testing whether the actual
proportion of homes with
high lead content in their
water ______________ 10%.
Two-sided test: when you’re testing an alternative
in ___________________________ (𝐻𝐴 uses ≠)
p-value for two-sided test: the probability of
getting results as different or more different from
𝐻0 in __________________, assuming 𝐻0 is true.
p-value for one-sided test: the probability of
getting results as different or more different from
𝐻0 in __________________, assuming 𝐻0 is true.
Calculate the two-sided p-value (show your work):
One-sided p-value = 0.0002
381
Two-sided test calculations:
To find the p-value for a two-sided test, just multiply the one-sided p-value _____!
One-sided conclusion:
Two-sided conclusion:
(p-value)
(α)
0.0002 < 0.05
(p-value)
(α)
_______ < 0.05
In both cases,
______________
In fields like medicine, two-sided tests are preferred because their ______________ p-values mean they
are more _________________—you are less likely to get a statistically significant result by chance alone.
Discussion
The confidence interval for the proportion of Flint homes with a high lead level in their water is
(0.121, 0.213).
Discussion Question: Is this confidence interval consistent with the results of our hypothesis test?
Explain.
382
Can you taste the rainbow?
Many students claim that they can taste the different colors of Skittles. Today we will conduct
an experiment and perform a significance test to see if students really can “taste the rainbow”.
Collect data:
How many correct? ______
How many total? ______
383
Significance Test for a Proportion
Important ideas:
Check Your Understanding
1. Sometimes parents and grandparents like to recount how difficult life was when they were kids, such as having to
walk 10+ miles to school (in the snow, uphill both ways). A random sample of 180 teenagers were selected and
40% had heard stories from their parents or grandparents about how difficult life was when they were kids. Do these
data provide convincing evidence at the α = 0.05 significance level that the proportion of all teenagers who have
heard stories from their parents or grandparents about how difficult life was when they were kids differs from 0.50?
2. A 95% confidence interval for the proportion of all teenagers who have heard stories from their parents or
grandparents about how difficult life was when they were kids is (0.328, 0.472). Explain how the confidence interval
is consistent with, but gives more information than, the test.
384
Is Yawning Contagious?
Mythbusters investigated this question. Here’s a brief recap. Each subject was placed in a
booth for an extended period of time and monitored by hidden camera. 34 subjects were
given a “yawn seed” by one of the experimenters: that is, the experimenter yawned in the
subject’s presence before leaving the room. The remaining 16 subjects were given no yawn
seed.
1. Draw an outline of Mythbuster’s experiment.
50 subjects
2. Here are the Mythbusters results.
Subject Yawned?
Yawn seed?
Yes
No
Yes
10
24
No
4
12
Total
14
36
Total
34
16
50
Call p1 the true proportion of yes yawn seed people who yawn. p̂1 = ________
Call p2 the true proportion of no yawn seed people who yawn. p̂ 2 = ________
What
is the
p̂1 - p̂ 2? ___________
he front
of difference
the room. in
Is proportions
it a fair deck?
3. Do the data provide some evidence that yawning is contagious? Why?
4. Adam Savage and Jamie Hyneman, the cohosts of Mythbusters used these data to
conclude that yawning is contagious. Do you agree?
385
In this Activity, your class will investigate whether the results of the experiment are
statistically significant OR if they could have occurred purely by chance due to random
assignment.
4. What is the null hypothesis?
The 50 people in the experiment are represented by the cards. A person is either a yawner
or a non-yawner, no matter which treatment they are randomly assigned.
5. Shuffle the 50 cards and put them into two piles, one group of 34 that gets the yawn
seed and one group of 16 that does not get the yawn seed. Record the proportion of
people who yawned in each group. You will do this three times.
Trial
Proportion who yawned
in yawn seed group, p̂1
Proportion who yawned
no yawn seed group, p̂ 2
Difference in
proportions, p̂1 - p̂ 2
1
2
3
6. Make a class dotplot of the difference in proportions. Sketch below:
7. In what percent of the class’s trials did the difference in proportions equal or exceed
29% - 25% = 4% (what Mythbusters got in their experiment)?
8. What conclusion can you make about whether yawning is contagious?
386
Tests About a Difference in Proportions - Intro
Important ideas:
Check Your Understanding
Mythbusters investigated the question “Is Yawning Contagious?” Here’s a brief recap.
Each subject was placed in a booth for an extended period of time and monitored by
hidden camera. 34 subjects were given a “yawn seed” by one of the experimenters: that is,
the experimenter yawned in the subject’s presence before leaving the room. The
remaining 16 subjects were given no yawn seed.
Yawn seed?
Yes
No
Total
Subject Yawned?
Yes
No
10
24
4
12
14
36
Total
34
16
50
387
Race and Hiring
Your class just performed a modified version of the famous race/resumé study* conducted by
researchers at the University of Chicago. You will now conduct a two-sample z-test for
proportions with the resulting data.
*Bertrand, Marianne and Sendhil Mullainathan. "Are Emily And Greg More Employable Than Lakisha And Jamal? A Field
Experiment On Labor Market Discrimination," American Economic Review, 2004, v94(4,Sep), 991-1013.
https://www.nber.org/papers/w9873
Surprise! The resumé you evaluated was fake. You just took part in an experiment. Each resumé the
class evaluated was completely identical except for one component: the first name of the applicant.
Half of the class got “Emily Jones,” and the rest got “Lakisha Jones.” According to birth certificate
records, the name “Emily” is almost exclusively given to white children. The name “Lakisha” is
almost exclusively given to black children.
Fill in the following based on data from your whole class:
Number of callbacks for Emily: _______
Number of callbacks for Lakisha: _______
Sample size for Emily group: _______
Sample size for Lakisha group: ______
Callback proportion for Emily: ______
Callback proportion for Lakisha: _______
1. Without performing any further calculations, do you think the difference in callback proportions
between the Emily and Lakisha resumés is significant? Why or why not?
2. Is there convincing evidence of a difference in callback rates, based on the name alone?
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
388
3. In the original study, the researchers sent resumés with commonly-white or commonly-black
names (randomly assigned) to firms in Boston and Chicago. In total, 246 out of 2445 commonlywhite named resumés received a callback and 164 out of 2445 commonly-black named resumés
received a callback. Is this evidence more convincing or less convincing of racial bias than the data
from our class experiment? Explain. Don’t perform another full test!
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
389
Tests About a Difference in Proportions
Important ideas:
Check Your Understanding
To study the long-term effects of preschool programs for poor children, researchers designed an
experiment. They recruited 123 children who had never attended preschool from low-income
families in Michigan. Researchers randomly assigned 62 of the children to attend preschool (paid for
by the study budget) and the other 61 to serve as a control group who would not go to preschool.
One response variable of interest was the need for social services as adults. Over a 10-year period,
38 children in the preschool group and 49 in the control group have needed social services.
Do these data provide convincing evidence that preschool reduces the later need for social services
for children like the ones in this study? Justify your answer.
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
390
AP Statistics Handout: Lesson 9.3
Topics: two-sample z-test for a difference of proportions and its connection to intervals
Guided Notes
Context for Today’s Lesson: The Race/Resumé Study
In this study*, investigators created mock identical resumés, which were sent to job placement ads in
Chicago and Boston. Each resumé was randomly assigned either a commonly-white or commonly-black
name. In total, 246 out of 2445 commonly-white named resumés received a callback and 164 out of
2445 commonly-black named resumés received a callback.
*Bertrand, Marianne and Sendhil Mullainathan. "Are Emily And Greg More Employable Than Lakisha
And Jamal? A Field Experiment On Labor Market Discrimination," American Economic Review, 2004,
v94(4,Sep), 991-1013. https://www.nber.org/papers/w9873
Two Sample Z-Test for a Difference of Proportions
Study results
CommonlyWhite Names
CommonlyBlack Names
Total
Called back
246
164
410
Not called back
2199
2281
4480
Total
2445
2445
4890
Find the following quantities (show
calculations): 𝑛1 , 𝑛2 , 𝑝̂1, 𝑝̂ 2 , 𝑝̂1 − 𝑝̂ 2
̂𝟏 > 𝒑
̂𝟐
If there’s hiring discrimination, 𝒑
Group 1: White
𝑝̂1 = proportion of commonly-white
name apps that got callback.
𝑝̂1 = _________
>
Group 2: Black
𝑝̂2 = proportion of commonly-white
name apps that got callback.
𝑝̂2 = _________
Are these proportions _____________________________ to show discrimination, or could this
difference have been a result of _________________________?
391
Hypotheses: Recall court analogy: “Innocent until proven guilty”
There is no discrimination,
so the callback rate is the
_______________________
. You’re seeing if there’s
evidence to reject this
default claim.
Non-numerical version:
Numerical version:
𝐻0 : Employers __________________
𝐻0 :
• Default/dull belief (___________________)
𝐻𝐴 : Employers ____________ white applicants
𝐻𝐴 :
• Unproven research hypothesis we aim to
demonstrate (_______________________)
Where:
𝑝1 is the proportion of ______ applicants with commonly-white names
who’d receive callbacks when applying to jobs like the ones in this study.
𝑝2 is the proportion of ______ applicants with commonly-black names
who’d receive callbacks when applying to jobs like the ones in this study.
There is discrimination, in
which case the commonlywhite named applications
received a ______________
of callbacks.
Rewrite these hypotheses in a more mathematically convenient way:
Under the hood: calculating z-test statistics and p-values for a two-sample test
Investigating if we find enough statistical evidence to reject the
𝑯𝟎 and support the 𝑯𝑨
𝐻0 : 𝑝1 − 𝑝2 = 0
𝐻𝐴 : 𝑝1 − 𝑝2 > 0
1. Start by assuming the null (the default) is true.
• We assume there is no discrimination: callback rates are the same for both groups.
2. Ask yourself: if the null is in fact true, how likely is the
data that you’ve gathered?
2. What estimates for 𝑝1 - 𝑝2 are we _________
_______ if there’s actually no discrimination?
Let’s find the ________________________ of
𝑝̂1 - 𝑝̂2 from all possible random assignments.
Under certain conditions:
𝑝̂𝑐 (1−𝑝̂𝑐 )
~ Norm(𝜇 = 0, 𝜎 = √
𝑛1
+
𝑝̂𝑐 (1−𝑝̂𝑐 )
~ Normal(𝜇𝑝̂ = 0, 𝜎𝑝̂ = _______)
392
𝑛2
Now we look at the data we gathered (difference in callback rates from experiment): 𝑝̂1 − 𝑝̂ 2 = 𝟎. 𝟎𝟑𝟒
Under the null assumption, how unlikely
was the data we observed (𝑝̂1 − 𝑝̂2 = 𝟎. 𝟎𝟑𝟒)?
Two ways to measure that:
1. z-test statistic
2. p-value
Write down the z-test statistic and the p-value. Interpret their meaning and describe if they suggest
there is reason to doubt the null hypothesis in this case.
3. Draw a conclusion:
Under my assumption that there is no difference in callback rates, the actually observed data (a 3.4%
difference in callback rates among 4890 employers) is highly unlikely (__________________). So, _____
___________________________. There’s convincing evidence that commonly-white named resumés
receive a _________________________.
Connection to Confidence Intervals
Confidence Interval:
(1.85%, 4.95%)
Since 𝑝1 − 𝑝2 = 0 was ________________ our interval, it’s not plausible to assume 𝑝1 = 𝑝2 . Therefore,
we ______________. There’s evidence of a preference for “white name” resumés.
Usually (especially for two-sided tests): If 𝐻0 is outside the interval, you will reject it. If 𝐻0 is __________
the interval, you _________________________.
393
In the Bertrand-Mullainathan race/resumé study, mock identical resumés were sent to job placement ads
in Chicago and Boston. Each resumé was randomly assigned either a commonly-white or commonlyblack name. In total, 246 out of 2445 commonly-white named resumés received a callback and 164 out
of 2445 commonly-black named resumés received a callback.
Do the results give convincing statistical evidence that employers favored commonly-white name
applicants (in terms of callbacks)?
Calculator steps for two-sample z-interval for proportions:
For groups 1&2:
x: # of ‘successes’
n: sample size
p1: 𝐻𝐴 direction
Calculate
STAT → TESTS →
6: 2-PropZTest…
394
Output
Lesson 9.3 Discussion
CommonlyWhite Names
CommonlyBlack Names
Total
Called back
246
164
410
Not called back
2199
2281
4480
Total
2445
2445
4890
Discussion Question: We found that commonly-white name resumés had a statistically significant
advantage in attracting callbacks. Does this result suggest discrimination at other levels of the hiring
process (e.g. job offers rates after an interview, starting salary estimates, etc.)? Explain.
395
Should Rockford switch to bottled water?
The Wolverine Worldwide (a shoe company in Rockford) improperly disposed of chemicals
(PFAS), which have leaked into the ground water. The state of Michigan says that if more
than 7% of households in a city exceed the safe limit, the city needs to switch to bottled water.
A concerned citizen takes a random sample of 100 households and finds that 12 have unsafe
water. Do the data provide convincing evidence that Rockford should switch to bottled water?
1. State appropriate hypotheses for performing a significance test. Use α = 0.05.
2. (a) After conducting a significance test, a P-value of 0.025 is found. Interpret this value.
(b) Reject H0 or fail to reject H0? Keep the current water or switch to bottled water? Explain.
(c) Let’s suppose this decision is wrong. What would be a consequence of this error?
(d) If the water is safe, what is the probability that this error will occur?
3. (a) Now suppose the P-value was 0.217. Reject H0 or fail to reject H0? Keep the current water
or switch to bottled water? Explain.
(b) Let’s suppose this decision is wrong. What would be a consequence of this error?
4. Are the consequences in question #2 or question #3 more serious? Explain.
396
Type I and Type II Errors
Important ideas:
Check Your Understanding
Mr. Wilcox purchased a trick coin that is supposed to land heads up 75% of the time. One of his
students volunteer to test this claim. The student flips the coin 50 times and finds that the coin lands
heads up 35 times. The student then performs a test of the following hypotheses at the α = 0.10
significance level:
𝐻! : 𝑝 = 0.75
𝐻" : 𝑝 < 0.75
where p = the true proportion of tosses of this coin that would land heads-up.
1. Describe a Type I error and a Type II error in this setting.
2. Which type of error may result in Mr. Wilcox returning the coin and writing a negative review
of the product?
3. If the student were to use α = 0.05 instead of α = 0.10, would this make it more or less likely to
reject the null hypothesis when the null hypothesis is true? Explain.
397
Will Mrs. Magic Prove Herself??
Recall that Mrs. Magic claims she is an 80% free throw
shooter. Mr. Wilcox is skeptical and believes that she is a
less than 80% free throw shooter. Mrs. Magic has decided to
take another SRS of 50 free throws to prove herself.
1. Write the appropriate hypotheses for the significance test. Be sure to define the
parameter of interest.
2. Describe a Type II error in this setting.
3. Suppose that Mrs. Magic s true free throw percentage is actually 66% (Ha is true!).
Enter all information into the applet at https://istats.shinyapps.io/power/. Click to
Display Type II error. It will show as a blue shaded region.
(a) If Ha is true, what is the probability of a Type II error (bad decision)? _______
(b) If Ha is true, what would be the probability of a good decision here? _______
(c) Interpret the probability of this good decision in the context of the problem.
4. Click Display Power. We want to increase the power of our test. How could we adjust
each of the following factors to increase our power? Use the applet to explore each.
a. Sample size:
b. Alpha level:
c. Alternative p:
398
Power of a Test
Important ideas:
Check Your Understanding
A statistics major and a finance major decide to get married. In order to investigate their per person
expenses they select a random sample guests they have invited to the wedding and records
whether or not each person plans to attend the wedding. They decide to test H0: p = 0.75 versus Ha: p
≠ 0.75 where p = the true proportion of all guests that will attend the wedding.
a. The power of the test to reject H0: p = 0.75 when p = 0.70 using α = 0.05 and n = 25 subjects is
0.10. Interpret this value.
b. Find the probability of a Type I error and the probability of a Type II error for the test in part (a).
c. Determine whether each of the following changes would increase or decrease the power of the
test. Explain your answers.
•
Use α = 0.01 instead of α = 0.05.
•
Use n = 100 instead of n = 25.
399
400
Formulas for Significance Tests
Formulas for Confidence
Intervals
Formula for standard deviation
of the sampling distribution
Formula for mean of the
sampling distribution
Name the procedure
Conditions
Symbol for statistic (sample)
Symbol for parameter
(population)
Lesson
One Proportion
AP Stats Chapter 8,9 Study Sheet
Difference of Proportions
®
2015 AP STATISTICS FREE-RESPONSE QUESTIONS
4. A researcher conducted a medical study to investigate whether taking a low-dose aspirin reduces the chance
of developing colon cancer. As part of the study, 1,000 adult volunteers were randomly assigned to one of two
groups. Half of the volunteers were assigned to the experimental group that took a low-dose aspirin each day,
and the other half were assigned to the control group that took a placebo each day. At the end of six years,
15 of the people who took the low-dose aspirin had developed colon cancer and 26 of the people who took
the placebo had developed colon cancer. At the significance level a = 0.05, do the data provide convincing
statistical evidence that taking a low-dose aspirin each day would reduce the chance of developing colon cancer
among all people similar to the volunteers?
401
2012 AP ® STATISTICS FREE-RESPONSE QUESTIONS
5. A recent report stated that less than 35 percent of the adult residents in a certain city will be able to pass a
physical fitness test. Consequently, the city's Recreation Department is trying to convince the City Council to
fund more physical fitness programs. The council is facing budget constraints and is skeptical of the report. The
council will fund more physical fitness programs only if the Recreation Department can provide convincing
evidence that the report is true.
The Recreation Department plans to collect data from a sample of 185 adult residents in the city. A test of
significance will be conducted at a significance level of a = 0.05 for the following hypotheses.
H 0 : p = 0.35
H a : p < 0.35,
where p is the proportion of adult residents in the city who are able to pass the physical fitness test.
(a) Describe what a Type II error would be in the context of the study, and also describe a consequence of
making this type of error.
(b) The Recreation Department recruits 185 adult residents who volunteer to take the physical fitness test. The
test is passed by 77 of the 185 volunteers, resulting in a p-value of 0.97 for the hypotheses stated above. If it
was reasonable to conduct a test of significance for the hypotheses stated above using the data collected
from the 185 volunteers, what would the p-value of 0.97 lead you to conclude?
(c) Describe the primary flaw in the study described in part (b), and explain why it is a concern.
402
Unit 7
Chapter 10
Estimating Means
with Confidence
403
AP Statistics Handout: Lesson 10.1
Topics: review sampling distribution for 𝑥̅, 95% confidence interval for 𝑥̅, four step process
Guided Notes
Students: “I’m gonna drop out and become a…”
YouTube Creator
Insta Influencer
“TikToker”
Mr. Beast
2020: $24 Million
Kendall Jenner
2019: $16 Million
Josh Richards
2020: $1.5 Million
Source: Forbes
Can you make this kind of money by making social media your full-time career? Probably not, but
maybe you could make a living…
Today’s Key Analysis: Do social media creators, on average, make a livable wage?
The Data:
Searched “How much I make on YouTube.” ___________________________ from
hundreds of results
Reliable data: They show their
private channel revenue pages in the
videos
Source:
Forbes
404
Review: Sampling Distribution for 𝑥̅
The Population vs. The Sample Mean
𝜇 = _________________ mean
• Parameter
• Ex: mean salary _____________________
𝑥̅ = _____________ mean
• Statistic used to estimate µ
• Ex: mean salary ________________
Estimation
In a world where…
1. The true mean yearly salary among YouTubers is $55,000
2. The true standard deviation of salaries is $29,500
We ________________ the
true mean by $5,000
In most real-life scenarios:
-You _______________________ the true mean.
-You survey a random sample of 35 YouTubers. Among them, the average yearly earnings was
𝑥̅ = $60,000.
-Your estimate for the true mean (µ) is the sample mean (𝑥̅ )
The Sampling Distribution
Under certain conditions:
𝑥̅ ~ Normal(𝜇𝑥̅ = 𝜇, 𝜎𝑥̅ =
Centered at true mean (______________)
𝜎
√𝑛
)
____________________________ of our
estimates from true mean value
Calculations:
𝑥̅ ~ Normal(𝜇𝑥̅ = 𝜇, 𝜎𝑥̅ =
𝜎
)
√𝑛
𝑥̅ ~ Normal(𝜇𝑥̅ = 55000, 𝜎𝑥̅ =
Means from repeated
random samples of 35
YouTubers (in a world
where µ = $55,000).
29500
)
√35
𝑥̅ ~ Normal(𝜇𝑥̅ = 55000, 𝜎𝑥̅ = 4986)
Typical distance of estimates from true mean value.
________________by $4,986.
405
We overestimated the
true mean by $5,000
1) Was our estimate’s error amount ($5,000) pretty typical? Justify using the sampling distribution
above.
There should be a way to report both:
a) Our estimate
b) How far off this estimate might be
One way to do this is with a _________________________________!
The t-distribution and interval for 𝑥̅
Sampled YouTuber Yearly Revenues ($)
1) Label the U.S. individual poverty line, the U.S. mean wage, the sample mean, and the
sample standard deviation on the dotplot above.
2) What concerns might you have about assuming the true mean matches the sample
mean above? Why?
406
First Attempt: Creating the Confidence Interval for our Data
Under certain conditions:
𝑥̅ ~ Normal(𝜇𝑥̅ = 𝜇, 𝜎𝑥̅ =
𝜎
√𝑛
)
1) What prevents us from directly calculating 𝜎𝑥̅ ? What can we do to provide a reasonable estimate?
2) Using the YouTube data, what is 𝑆𝐸𝑥̅ ? Show your work.
3) Why can’t we directly calculate the confidence interval
using the method show on the diagram to the right?
Note: we can’t do this to calculate the interval
The t-distribution to the Rescue!
t distribution
• ___________ than normal curve
• Width captures ___________________________
when using 𝑆𝑥 to estimate σ
degrees of freedom (df) = _______
• df measures how _______________ 𝑆𝑥 estimates σ
• The higher n, the more precisely we estimate σ, and
the more our t-curve ________________________!
407
t critical values
𝑡 ∗ : the _____________________ of the t-interval
• Tells you how many _____________________ you’re including in your interval.
• Determines the _____________________.
Normal (z)
t (df = 34)
For normal curve, z* was
1.96 for 95% confidence
For t-distribution, we need
to go wider to capture 95%
t (df = 34)
Calculator steps to find t*
t* = 2.03
2nd → VARS → 4: InvT
-Area: percent below
interval (2.5% = 0.025)
-df: n – 1 (35 – 1 = 34)
As expected, the t-critical value is __________________ than 1.96 needed for a normal curve.
Confidence Interval for a Mean
Using the formula and the info above, calculate and interpret
the confidence interval for the mean YouTube salary:
408
Formula:
• point estimate ± margin of error
• 𝑥̅ ± 𝑡 ∗ (𝑆𝐸𝑥̅ )
a) Construct and interpret a 95% confidence interval for the true mean salary of all YouTubers.
Calculator steps
→Stats
𝑥̅ : sample mean
𝑆𝑥 : sample stdev
n: sample size
Calculate
STAT → TESTS →
8: TInterval
409
Output
Lesson 10.1 Discussion
We can directly estimate the true mean income of YouTubers from Ad Revenue:
/
Creator Share of 2019 YouTube Ad Revenue
$8,331,950,000
# of Creators
18,000,000
=
Average Yearly Pay Per Creator
$463
Confidence interval we estimated: $41,877 to $98,339
Discussion Question: Why was our confidence interval so far off?
Sources:
- Alphabet 4th quarter earnings release: https://abc.xyz/investor/static/pdf/2019Q4_alphabet_earnings_release.pdf?cache=05bd9fe
- “How YouTube Ad Revenue Works,” Investopedia, June 4, 2020: https://www.investopedia.com/articles/personal-finance/032615/howyoutube-ad-revenue-works.asp#:~:text=Enabling%20ads%20on%20your%20YouTube,get%20the%20remaining%2055%20percent.
- “How Many YouTube Channels Are There?”, Tubics, https://www.tubics.com/blog/number-of-youtube-channels
- SocialBlade, data accessed 2019
410
10.1 PRACTICE!
How much homework?
Ms. Garcia wants to estimate how much time students spend on homework, on average, during a typical week.
She wants to estimate at the 90% confidence level with a margin of error of at most 30 minutes. A pilot study
indicated that the standard deviation of time spent on homework per week is about 154 minutes.
SAT Math Scores
High school students who take the SAT Math exam a second time generally score higher than on their first try.
Past data suggest that the score increase has a standard deviation of about 50 points. How large a sample of
high school students would be needed to estimate the mean change in SAT score to within 2 points with 95%
confidence? Show your work.
Travel Time to Work
A study of commuting times reports the travel times to work of a random sample of 20 employed adults in New
York State. The mean (𝑥𝑥̅ ) is 31.25 minutes, and the standard deviation is (𝑠𝑠) 21.88 minutes.
What is the standard error of the mean? Interpret this value in context.
411
Vitamin C Content
Several years ago, the U.S. Agency for International Development provided 238,300 metric tons of corn- soy
blend (CSB) for emergency relief in countries throughout the world. CSB is a highly nutritious, low- cost fortified
food. As part of a study to evaluate appropriate vitamin C levels in this food, measurements were taken on
samples of CSB produced in a factory.
The following data are the amounts of vitamin C, measured in milligrams per 100 grams (mg/100 g) of blend, for
a random sample of size 8 from one production run:
26
31
23
22
11
22
14
31
Construct and interpret a 95% confidence interval for the mean amount of vitamin C (𝜇𝜇) in the CSB from this
production run.
412
How much does an Oreo weigh?
Mrs. Gallas wanted to estimate the average weight of an Oreo cookie to determine if the
average weight was less than advertised. She selected a random sample of 30 cookies and
found the weight of each cookie (in grams). The mean weight was x = 11.1921 grams with a
standard deviation of sx = 0.0817 grams. Make a 95% confidence interval to estimate the true
mean weight of an Oreo.
1. What is the point estimate for the true mean? ________
2. Identify the population, parameter, sample and statistic.
Population:
Parameter:
Sample:
Statistic:
3. Was the sample a random sample? Why is this important?
4. What is the formula for calculating the standard deviation of the sampling distribution of 𝑥̅ ?
5. What condition must be met to use this formula? Has it been met?
6. In the formula for standard deviation of the sampling distribution of 𝑥̅ , we don’t know the value
of σ (if we did, we would have known µ) so we will use 𝑠! instead. Find the standard error.
7. Would it be appropriate to use a normal distribution to model the sampling distribution of 𝑥̅ ?
Justify your answer.
8. When finding the margin of error for a confidence interval for a proportion we use z*. For a
mean we will use _____ as the critical value. Why???
413
Name:
Hour:
Date:
9. What t* is needed for this confidence interval? Use Table B and the degrees of freedom
(df) = n - 1 to find it.
10. Calculate the margin of error using t* and the standard error.
11. Calculate the 95% confidence interval using point estimate +/- margin of error.
12. Interpret the interval.
13. Write a specific formula for a confidence interval for a population mean.
14. According to Nabisco, an Oreo weighs 11.3 grams. Does our confidence interval provide
convincing evidence that the true average weight is less than 11.3 grams? Explain.
414
Estimating a Population Mean Day 1
Important ideas:
Check Your Understanding
1. Use Table B to find the critical value t* that you would use for a confidence interval for a
population mean μ in each of the following settings. If possible, check your answer with
technology.
(a) A 98% confidence interval based on a random sample of 26 observations
(b) A 99% confidence interval from an SRS of 85 observations
2. A national poll of a random sample of 1,640 adults was carried out by Morning Consult. Each
person reported how much they (or their significant other) spent on an engagement ring. The
histogram displays the results. Determine if the conditions for constructing a confidence
interval for a mean have been met in this context.
415
How many states can you name?
How many states can you name in one minute? We will use this class as a random sample
of all AP Stats students to estimate a 95% confidence interval for the mean number of
states an AP Stats student can name in one minute.
1. When the timer starts, list as many states as you can on a piece of paper. Write the
number of states you listed on the board.
2. What type of data is this? Categorical or quantitative?
2. Enter the class data at stapplet.com. Find the sample mean and standard deviation. Sketch
the dotplot of the sample data.
n=
x=
sx =
3. Construct a 95% confidence interval to estimate the mean # of states a senior can name.
416
Important ideas:
Check Your Understanding
City council members want to estimate how many pounds of trash households in their community
produce per week. To determine an estimate for the standard deviation of the weight of trash produced
a small random sample of households was selected and their trash was weighed on garbage day. This
produced an estimated standard deviation of 36 pounds.
a. How many households need to be surveyed to estimate µ at the 95% confidence level with a margin
of error of at most 3 pounds?
b. After solving part (a), the city council realizes that it would be too much work to weigh the garbage
of that many households. They give up on their hopes of estimating the true mean weight of trash
produced within 3 pounds and select a random sample of 15 households and weigh the trash for
each of these households on garbage day. Here are the results:
114.8
74.3 80.1 41.5 99.1 31.0 93.1 118.9 26.5 33.1 88.3
46.1 119.7 46.4 19.8
Calculate and interpret a 95% confidence interval for the mean weight of trash for all households in
this community.
417
AP Statistics Handout: Lesson 10.2
Topics: two-sample t-interval for a difference of means
Guided Notes
Two Facts about College:
1. On average, college graduates make ___________________________ than high school
graduates. ($30,000 more per year, in 2018)
2. ________________________________ of college, the average economic return to a college
degree is still _____________________________________________.
Are college and income causally related?
Causation
Correlation
For “margin” students (those on the
fence of attending college based on
motivation and qualifications)…
• If college effect is causal: Educators
should encourage margin students
to attend college
• If college effect is correlation:
Educators shouldn’t always
encourage margin students to
attend college
Florida College Study*
A Yale economist used a regression discontinuity (a “quasi-experimental” method) to explore whether
college has a causal impact on wages. AP Stats doesn’t cover quasi-experimental methods, but they’re
really cool. We’ll analyze a modified version of this study using a concept we do learn in AP Stats: the
two-sample t-interval. In the space below, describe how the Florida College Study obtained its two
comparison samples. Then, describe why their methods allow for a causal comparison:
*Study: Zimmerman, Seth. “The Returns to College Admission for
Academically Marginal Students.” Journal of Labor Economics, 2014,
vol. 32 (4). https://www.jstor.org/stable/10.1086/676661?seq=1
418
Technical notes for technical folks:
• Regression Discontinuity uses multiple regression, which isn’t covered in AP Stats. We’re doing a
modified version with a two-sample t-interval (which is covered in AP Stats)
• Technically, we’re not really doing regression discontinuity. But what we’re doing uses similar
logic and reflects the paper’s main outcomes.
• We modified the data to make sure our results (including standard errors) would be similar to
Zimmerman’s (specifically, his IV effect estimates for students with GPA’s within 0.15 points of
the cutoff - see Table 5 in his paper – using annual rather than quarterly earnings as the
outcome).
Study results, for students near the GPA cutoff:
Mean Income
Stdev. Income
n
College
$35,764
$27,147
202
High School
$28,964
$21,899
190
“Income” is students’ annual income 8-14
years after high school (in 2005 dollars)
If college made a difference for margin
students, just-above cutoff students should
have a ________________________________
than just-below cutoff students.
Show calculations for 𝑥̅𝐶 − 𝑥̅𝐻𝑆 :
Could the difference in outcome have happened by chance alone? Or did college admission actually
raise wages? Let’s make a ____________________________ to see if difference between these means is
significantly large.
Two-sample t-interval for a difference of means
Hypotheses: We don’t need to set up hypotheses to construct a confidence interval, but doing so is going
to help us conceptualize why our interval is useful.
𝐻0 : 𝜇𝐶 = 𝜇𝐻𝑆
𝐻𝐴 : 𝜇𝐶 > 𝜇𝐻𝑆
The null (______________) hypothesis: there is ___________________
in average wage between the college and high school groups. You’re
seeing if there’s evidence to reject this claim.
The alternative (________________) hypothesis: the college group
_______________________ (on average) than the high school group
Rewrite these hypotheses in a more mathematically convenient way:
Where:
μC is the mean earnings among _____ college students (just above GPA cutoff)
μHS is the mean earnings among _____ high school students (just below GPA cutoff)
419
Making the interval
Mean Income
Stdev. Income
n
College
$35,764
$27,147
202
High School
$28,964
$21,899
190
Under certain conditions:
𝜎 2
~ Norm(𝜇 = 𝜇𝐶 − 𝜇𝐻𝑆 , 𝜎 = √ 𝑛𝑐 +
𝑐
𝜎𝐻𝑆 2
𝑛𝐻𝑆
1. Show the steps we take to get our t-distribution (and its final parameters):
2. Show the steps for calculating the final confidence interval (and write down the interval):
3. Interpret your final interval and comment on whether 0 is inside or outside your interval (and why
that’s important):
420
)
In the Florida College Study, two “as good as randomly” assigned groups of students (with similar GPA’s)
were given admission to college or were denied. Summary statistics are provided.
a) Construct and interpret a 95% confidence interval for the difference in mean incomes between
marginal students who attend and do not attend college.
b) Use your interval to draw a conclusion about whether college creates an “income boost” for marginal
students.
Calculator steps for two-sample t-interval for means:
STAT → TESTS →
0: 2-SampTInt…
421
For groups 1&2:
𝑥̅ : sample mean
𝑠𝑥 : sample stdev.
n: sample size
C-lvl: Confidence Lvl
Pooled → NO
Output
Lesson 10.2 Discussion
Confidence interval: $1,906 to $11,693
Interval shows us that the “college boost” is statistically significant, but is it practically
• Study author (Zimmerman) shows, even with rising college costs, this “college boost” represents
important?
a substantial net gain over time.
• So, yes, it’s practically important! College “pays off” for students at the margin of admission.
Policy Implications
Many colleges have tried to recruit more students from underrepresented backgrounds (e.g. students of
color and low-income students).
• One problem: Because students from these backgrounds disproportionately attend lowerperforming schools, many have lower GPA’s and academic qualifications.
• Colleges worry underqualified students will have bad outcomes (not graduate, student debt,
etc.)
Discussion: To help recruit students from a wider array of backgrounds, Universities in Florida are
considering lowering their admissions standard for high school GPA’s. Would this be a wise move?
Explain your answer using the Florida College Study results.
422
10.2 PRACTICE!
Plastic Grocery Bags
Do plastic bags from Target or plastic bags from Publix hold more weight? A group of AP Statistic students
decided to investigate by filling a random sample of 5 bags from each store with common grocery items until the
bags ripped. Then they weighed the contents of items in each bag to determine its capacity. Here are their
results, in grams:
Target:
12,572
13,999
11,215
15,447
10,896
Publix:
9552
10,896
6983
8767
9972
Construct and interpret a 99% confidence interval for the difference in mean capacity of plastic grocery bags
from Target and Publix. Does your interval provide convincing evidence that there is a difference in the mean
capacity among the two stores?
423
Which cookie has the most chips?
Is there a difference in the number of chocolate chips in Chips Ahoy cookies versus the
number of chocolate chips in Meijer brand cookies? Each pair of students will count the
number of chocolate chips in 1 Chips Ahoy cookie and 1 Meijer Chipsters cookie. Due to
the factories processes, we can assume the population distributions of # of chips are
approximately normal and that the samples are random.
1. Record the number of chocolate chips in each cookie. Write them on the board.
# in Chips Ahoy = _______
# in Meijer Chipsters = _______
2. Find the mean number of chocolate chips for each type of cookie, the standard
deviation and the difference.
Chips Ahoy: x1 =
Meijer Chipsters: x2 =
s1 =
s2 =
Difference: x1 - x2 =
3. If we repeated this process many times and created a dotplot, we would have the
sampling distribution of x1 - x2. Describe the shape, center and spread of the sampling
distribution.
Shape:
Center:
Spread:
4. Have the conditions for constructing a confidence interval been met? Explain.
5. Construct a 95% confidence interval for the true difference in the mean number of
chocolate chips in Chips Ahoy and Meijer Chipsters.
6. Do we have evidence that there is a difference in the average number of chocolate
chips in a Chips Ahoy and a Meijer Chipsters cookie?
424
Confidence Intervals for a Difference in Means
Important ideas:
Check Your Understanding
The most recent American Time Use Survey, conducted by the Bureau of Labor Statistics, found
that many Americans barely spend any time reading for fun. People ages 15 to 19 average only 7.8
minutes of leisurely reading per day with a standard deviation of 5.4 minutes. However, people ages
75 and over read for an average of 43.8 minutes per day with a standard deviation of 35.5 minutes.
These results were based on random samples of 975 people ages 15 to 19 and 1050 people ages
75 and over.
Construct and interpret a 95% confidence interval for the difference in mean amount of time
(minutes) that people age 15 to 19 and people ages 75 and over read per day.
425
Does Memory Training Help?
Does memory training help? You will be given a list of words. You will be given 3 minutes to
memorize as many as possible by just rereading the words. You will record as many words as
you can remember. You will then be given another list of words to memorize for 3 minutes using a
memorization strategy. You will record as many words as you can remember.
1. How many words did you get correct using strategy 1 (rereading)?
2. How many words did you get correct using the strategy 2 (story)?
3. Add your data to the table on the board. Copy the data below and calculate the difference
in each pair.
Strategy 1
Strategy 2
Difference
(S2 – S1)
Strategy 1
Strategy 2
Difference
(S2 – S1)
Strategy 1
Strategy 2
Difference
(S2 – S1)
4. Use the 1 Quantitative Variable (single group) applet at stapplet.com to enter the
Differences only. Make a dotplot of the differences below:
What does the dotplot suggest about the memory training?
Mean:
Interpret:
SD:
426
5. Construct a 95% confidence interval for the true mean difference in words remembered by
students using story rather than rereading.
6. Do we have convincing evidence that there is more words remembered using a story
instead of rereading?
427
Confidence Interval for Paired Data
Important ideas:
Check Your Understanding
Teenagers spend, on average, approximately 5 hours online every day. Do parents realize how many
hours their children are spending online? A family psychologist conducted a study to find out. A random
sample of 10 teenagers were selected. Each teenager was given a Chromebook and free internet for 6
months. During this time their internet usage was measured (in hours per day). At the end of the 6
months, the parents of each teenager were asked how many hours per day they think their child spent
online during this time frame. Here are the results.
Teenager
Actual time spent online
(hours/day)
Parent perception
(hours/day)
Difference (A – P)
1
2
3
4
5
6
7
8
9
10
5.9
6.2
4.7
8.2
6.4
3.8
2.9
7.1
5.2
5.8
2.5
3
3
3.5
1.5
2
2
3
2.5
3
3.4
3.2
1.7
4.7
4.9
1.8
0.9
4.1
2.7
2.8
a. Make a dotplot of the difference (A – P) in time spent online (hours/day) for each teenager. What
does the dotplot reveal?
b. Find the mean and standard deviation of the difference (A – P) in time spent online. Interpret the
mean difference in context.
c. Construct and interpret a 90% confidence interval for the true mean difference (A – P) in time
spent online.
428
AP Statistics Handout: Lesson 10.2 part 2
Topics: matched-pairs t-interval and t-test
Guided Notes
Climate Change Skepticism:
97% of climate scientists:
current climate change is
real and caused by humans
3% of climate scientists and
author Michael Crichton
State of Fear: A popular book by science fiction author Michael Crichton –
who has an MD but did not pursue medicine. The book was a best-seller
and remains one of the most cited works by climate change skeptics.
The book is framed as fiction, but it has tons of graphs of real data,
footnotes to scientific articles, and a 20-page bibliography.
Senator James Inhofe, who used to chair the Senate committee on
Environment and Public Works, when asked about Crichton: "I think I've
read most of his books; I think I've read them all. I enjoyed most 'State of
Fear' and made it required reading for this committee."
Quote from: Jankofsky, M. “Michael Crichton, Novelist, Becomes Senate Witness.” The New York Times. Sept. 29, 2005.
https://www.nytimes.com/2005/09/29/books/michael-crichton-novelist-becomes-senate-witness.html
Identifying a matched-pairs situation
“It’s the record from the weather station
at Punta Arenas, near here. It’s the
closest city to Antarctica in the world.” He
tapped the chart and laughed. “There’s
your global warming.”
Punta Arenas
- from State of Fear
429
1. What is the general trend shown in the chart shown from the weather station at Punta Arenas? How
does this support Crichton’s skepticism of global warming?
2. Fill in the chart and complete the blank below:
Station
Punta Arenas
1901-1950 Temp
1951-2000 Temp
6.828 (oC)
Difference
6.752 (oC)
Temperatures tended to __________________ during the 1900s in Punta Arenas.
Selected Station
Sable Island
Manila Int Airport
Perm
Hobart Ellerslie
Bulawayo Goetz
Veraval
Yokohama
Punta Arenas
Aldergrove
Harare Kutsaga
Bahia Blanca Aero
Cape Leeuwin
Maliye Karmakuly
Hobarttasmanwas
Svyatoy
Apia
Aparri
Syktyvkar
Upernavik
Gabo Island
Antananarivoville
Kumasi
Khartoum
Mahe Seychellesbri
Onslow
Rarotonga Intl
Ponta Delgada
Viljujsk
Andenes
Kyzylorda
Port Blair
Chatham Islands
1901-1950 1951-2000 Difference
Temp
Temp
(oC)
6.803
7.420
0.617
26.779
27.416
0.637
1.670
2.208
0.538
12.549
13.062
0.513
18.891
19.183
0.292
26.404
26.779
0.375
14.534
15.428
0.894
6.828
6.752
-0.077
8.888
9.012
0.124
18.816
19.055
0.239
14.963
15.204
0.241
16.608
16.983
0.375
-5.036
-5.044
-0.008
12.439
12.638
0.199
-0.263
0.263
0.527
26.380
26.479
0.099
25.288
26.091
0.803
0.435
0.894
0.460
-7.012
-7.286
-0.274
14.925
14.924
-0.001
17.387
17.741
0.354
25.652
25.854
0.202
28.535
28.874
0.339
26.414
26.872
0.459
24.154
24.540
0.387
24.105
24.157
0.052
14.942
15.676
0.734
-8.854
-9.057
-0.203
2.600
3.160
0.560
9.631
10.411
0.780
26.506
26.778
0.271
10.392
11.132
0.739
Here’s the thing: Punta Arenas is one location. Climate
patterns vary a lot by location. Instead of one location,
let’s ask: What is the average trend across many of these
stations? We collected temperature readings from 32
NASA-GISS stations based on a random sample of
latitude-longitude coordinates.
This data has a _________________________ structure.
•
•
Each individual (station) has _________
____________________
We find differences _____________________
_______________ (within station)
The matched-pairs data structure is useful because we
are __________________________________ in weather
patterns between stations.
Data Note: Temperatures are averages of the meteorological annual
mean temperatures from those years (in Celsius). Data is collected
from the "Adjusted cleaned" dataset from NASA, which they describe
as, "adjusted data after removal of some outliers and duplicate
records.“ Crichton appears to use the unadjusted and uncleaned data
in the chart displayed in State of Fear, but the patterns at Punta
Arenas and other areas are largely the same.
Data Source:
https://data.giss.nasa.gov/gistemp/station_data_v4_globe/#form
430
The station data
Punta Arenas: -0.076 oC
Temperature Differences from Stations (Late 1900’s – Early 1900’s) in Celsius
1. Did more stations see warming during the 1900’s or cooling during the 1900’s? How do you know?
2. Does Punta Arenas appear to be representative of the general trend we see in the station data? How
does this influence your evaluation of Michael Crichton’s evidence against global warming?
Punta Arenas
3. What is the sample mean of the differences? Draw this on the dot plot.
4. Is the sample mean more representative of the
general trend than Punta Arenas? Why or why not?
Matched-pair t-interval
From: https://climate.nasa.gov/interactives/climate-time-machine
Thank you Amy Hogan (@alittlestats) for sharing!
The cool thing about inference for matched-pairs: It’s the same as one-sample inference! We’re finding
a ___________________ of the differences.
Summary statistics you’ll be working with: 𝑥̅𝐷 = 0.35 , 𝑠𝑥 = 0.296, n = 32
431
a) Construct and interpret a 95% confidence interval for the true global average difference in
temperature (Late 1900’s – Early 1900’s). Display your full proceess below:
For “DO” phase: Calculator steps
→Stats
𝑥̅ : sample mean
𝑆𝑥 : sample stdev
n: sample size
Calculate
STAT → TESTS →
8: TInterval
432
Output
Matched-pair t-test
b) Determine if we have statistically convincing evidence of true global warming (on average). One more
time, show the full process. You don’t have to check conditions again.
Calculator steps
STAT → TESTS →
2: T-Test
→Stats
𝜇0 : null mean
𝑥̅ : sample mean
𝑆𝑥 : sample stdev
> 𝜇0 : alternative
c) Is your confidence interval (part a) consistent with your answer to part b? Explain.
433
Output
Discussion
We’re often taught: “There are two sides to every story.” In popular media, climate change is often
framed as a debate between two sides:
News stories on climate often interview two experts:
Expert 1
Expert 2
“Climate change
is real and
human-caused.”
“Climate change is
not real or humancaused.”
97% of climate
scientists agree
3% of climate
scientists agree &
Discussion Question: Should popular media portray both sides of the climate change “debate?” Explain
your reasoning.
434
®
2013 AP STATISTICS FREE-RESPONSE QUESTIONS
STATISTICS
SECTION II
Part A
Questions 1-5
Spend about 65 minutes on this part of the exam.
Percent of Section II score-75
Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the
correctness of your methods as well as on the accuracy and completeness of your results and explanations.
1. An environmental group conducted a study to determine whether crows in a certain region were ingesting food
containing unhealthy levels of lead. A biologist classified lead levels greater than 6. 0 parts per million (ppm) as
unhealthy. The lead levels of a random sample of23 crows in the region were measured and recorded. The data
are shown in the stemplot below.
Lead Levels
2
3
3
4
4
5
5
6
6
8
0
588
112
688
0122 34
99
34
68
Key: 218 = 2.8 ppm
(a) What proportion of crows in the sample had lead levels that are classified by the biologist as unhealthy?
(b) The mean lead level of the23 crows in the sample was 4.90 ppm and the standard deviation was1.12 ppm.
Construct and interpret a95 percent confidence interval for the mean lead level of crows in the region.
435
2008 AP ® STATISTICS FREE-RESPONSE QUESTIONS
3. A local arcade is hosting a tournament in which contestants play an arcade game with possible scores ranging
from 0 to 20. The arcade has set up multiple game tables so that all contestants can play the game at the same
time; thus contestant scores are independent. Each contestant's score will be recorded as he or she finishes, and
the contestant with the highest score is the winner.
After practicing the game many times, Josephine, one of the contestants, has established the probability
distribution of her scores, shown in the table below.
Josephine's Distribution
16
17
18
19
0.10
0.30
0.40
0.20
Score
Probability
Crystal, another contestant, has also practiced many times. The probability distribution for her scores is shown in
the table below.
Crystal's Distribution
Score
Probability
17
18
19
0.45
0.40
0.15
(a) Calculate the expected score for each player.
(b) Suppose that Josephine scores 16 and Crystal scores 17. The difference (Josephine minus Crystal) of
their scores is -1. List all combinations of possible scores for Josephine and Crystal that will produce
a difference (Josephine minus Crystal) of -1, and calculate the probability for each combination.
(c) Find the probability that the difference (Josephine minus Crystal) in their scores is -1.
(d) The table below lists all the possible differences in the scores between Josephine and Crystal and some
associated probabilities.
Distribution (Josephine minus Crystal)
Difference
-3
Probability
0.015
-2
-1
0
1
2
0.325
0.260
0.090
Complete the table and calculate the probability that Crystal's score will be higher than Josephine's score.
436
Unit 7 Chapter
11 Testing
Claims About
Means
437
AP Statistics Handout: Lesson 11.1
Topics: one sample t-test for a mean
Guided Notes
SNAP benefits (food stamps) should cover the following essential monthly bundle of groceries for one
person:
• 4 boxes of cereal, 1 gallon of milk, 5 banana bunches, 2 gallons of
orange juice
• 3 loaves of bread, 4 packages of sliced turkey (9-oz each), 3
packages of sliced provolone (10 count), 4 packages of romaine
lettuce (7 oz each), 12 tomatoes, small mayo jar (15 oz), 30
apples, 4 bags of mixed nuts (8 oz each), 5 diet coke 6-packs.
• 15 lbs boneless chicken breasts, 6 packs of frozen mixed veggies
(16 oz each), 4lbs white rice, 10 cans of black beans
• Salt (5 oz), pepper (5 oz), olive oil (17 oz)
We’ll call this grocery list the essentials bundle. We will use sampling to determine if SNAP benefits
cover the average (mean) cost of this grocery bundle in U.S. grocery stores.
One Sample t-test for a Mean
Central question: In fiscal year 2020, the maximum monthly SNAP allotment for a household of one
person (who could not earn income) was $194 in the U.S.* An SRS of 31 U.S. grocery stores found the
average cost of the essentials bundle was $199.45, with a sample standard deviation of $25.98.
Is there convincing statistical evidence that the maximum SNAP individual monthly allotment ($194)
doesn’t cover the true mean price of the essentials-only grocery bundle in the U.S.?
How can we tell this question is asking us to do a hypothesis test?
*This is the max federal allotment for the continental U.S. (not Alaska/Hawaii). Source: U.S. Department of Agriculture, “SNAP
Fiscal Year 2020 Cost-of-Living Adjustments,” https://fnsprod.azureedge.net/sites/default/files/media/file/COLA%20Memo%20FY%202020.pdf
Building Hypotheses
Null hypothesis (𝐻0 ): The ________________ belief about a parameter’s value.
Alternative hypothesis (𝐻𝐴 ): Our unproven belief about a ____________________________, for which
we have to gather evidence.
438
Convert the following hypotheses into numerical statements:
Hypotheses in words
𝐻0 : Food stamps cover the true
average cost of essentials in U.S.
𝐻𝐴 : Food stamps don’t cover it
Hypotheses in numbers
𝐻0 :
𝐻𝐴 :
Let µ = true mean price of the essentials-only grocery bundle in the U.S.
Performing the Test, Step 1
Step 1: Start by assuming the null (the default) ________________.
We assume SNAP provides enough:
the true mean price of the essentialsonly bundle in U.S. stores is
_______________________________
______________________.
Performing the Test, Step 2
Step 2: Ask yourself: if the null is in fact true, how likely is the data that you’ve gathered?
We collected bundle prices from 31
random grocery stores (n = 31). At this
sample size, what estimates for µ are
we __________________? Let’s find
the ___________________________
________ of 𝑥̅ ’s from all possible
samples.
Under certain conditions:
𝑥̅ ~ 𝑁𝑜𝑟𝑚 (𝜇𝑥̅ = 𝜇, 𝜎𝑥̅ =
Now, we look at the data we sampled: among 31 stores,
̅ = $199.45
the mean price of bundle was: 𝒙
439
𝑥̅ ~ 𝑡 (df = 30)
𝜇𝑥̅ = ______
𝑆𝐸𝑥̅ = ______
𝜎
√𝑛
)
Using the t-test statistic:
Z-Scores
Shows how many σ a data value
is above/below the mean.
𝑧=
t-test statistics
Shows how many SE a _________________ is above / below _____,
using a t-dist.
data value − mean
standard deviation
z=
t=
𝑥̅ −𝜇0
𝑆𝐸𝑥̅
→ t=
𝑥̅ −𝜇0
𝑠𝑥
√𝑛
𝑥𝑖 − µ
𝜎
The _________________ the test statistic (in magnitude), the less ________________ your
measurements are under the null assumption. You can’t _________________________________.
𝑡=
𝑥̅ − 𝜇0
=
𝑆𝐸𝑥̅
𝑡=
Getting a sample mean about 1 standard
error away from the hypothesized mean is
_________________________.
Using the p-value: exactly how unlikely was the proportion we sampled?
a) This probability is the p-value. Explain why:
b) Show the calculator commands and final
result for the p-value:
Performing the Test, Step 3: Write the Conclusion
Under my assumption that the true mean price of the essentials bundle is $194 in U.S. stores, the actual
observed data ($199.45 sample mean at 31 stores) is somewhat likely (________________). So, ______
___________________________________. We don’t have convincing evidence that the average price of
the bundle is higher than SNAP benefits.
440
Is there convincing statistical evidence that SNAP ($194) doesn’t cover the average cost of essential
food? An SRS of 31 U.S. grocery stores found the average cost of the essentials bundle was $199.45,
with a sample standard deviation of $25.98.
Calculator steps for one-sample z-test for a proportion:
→Stats
μ0 : null value
x̅: sample mean
sx : sample stdev.
n: sample size
µ: HA direction
STAT → TESTS →
2: T-Test
441
Output
11.1 PRACTICE!
Anemia in Children
Hemoglobin is a protein in red blood cells that carries oxygen from the lungs to body tissues. People with less
than 12 grams of hemoglobin per deciliter of blood (g/dl) are anemic. A public health official in Jordan suspects
that Jordanian children are at risk of anemia. He measures a random sample of 50 children.
(a) State the appropriate null hypothesis Ho and alternative hypothesis Ha. Be sure to define your parameter.
For the study of Jordanian children, the sample mean hemoglobin level was 11.3 g/dl and the sample standard
deviation was 1.6 g/dl. A significance test yields a P-value of 0.0016.
(b) Interpret the p-value in context.
(c) What conclusion would you make if α = 0.05? α = 0.01? Justify your answer.
442
Healthy Streams
The level of dissolved oxygen (DO) in a stream or river is an important indicator of the water’s ability to support
aquatic life. A dissolved oxygen level below 5 mg/l puts aquatic life at risk. A researcher measures the DO level
at 15 randomly chosen locations along a stream. Here are the results in milligrams per liter:
4.53
5.04
3.29
5.23
4.13
5.50
4.83
5.42
6.38
4.01
4.66
2.87
5.73
5.55
Is there evidence that aquatic life is at risk in this stream?
443
4.40
Pineapples
At the Hawaii Pineapple Company, managers are interested in the sizes of the pineapples grown in the
company’s fields. Last year, the mean weight of the pineapples harvested from one large field was 31 ounces. A
new irrigation system was installed in this field after the growing season. Managers wonder whether this
change will affect the mean weight of future pineapples grown in the field. To find out, they select and weigh a
random sample of 50 pineapples from this year’s crop. The Minitab output below summarizes the data.
Is there evidence that the mean weight of pineapples is different with the new irrigation system?
444
Are you getting enough sleep?
It’s recommended that teenagers get 8 hours of sleep a night. Mrs. Gallas believes her AP
Stats students are getting less than the recommended 8 hours of sleep per night. To test
her belief, take a random sample of 10 students in class and record the number of hours
of sleep for each. Do these data provide convincing evidence that the AP stats students
get less than 8 hours of sleep per night using α = 0.05?
1. Calculate the sample mean and standard deviation.
2. State the appropriate hypotheses for a significance test. Be sure to define the
parameter of interest.
3. What conditions must be met? Check them.
4. Give the formulas for the mean and standard deviation of the sampling distribution of x
and calculate the values.
5. Draw a picture and then calculate the test statistic.
6. Remember, since we are working with means, the test statistic is a t value. Use table
B to find the P-value.
7. What conclusion can we make?
445
Significance Test for µ
Important ideas:
Check Your Understanding
According to the AAA Foundation for Traffic Safety’s American Driving Survey, U.S. drivers spend,
on average, 51 minutes behind the wheel each day. A researcher believes this is an overstatement.
To investigate, a random sample of 75 drivers were selected. The study revealed that the mean
time behind the wheel for the sample of 75 drivers was 46.4 minutes with a standard deviation of
18.8 minutes. Is there convincing evidence that the mean time behind the wheel for all U.S. drivers
is less than 51 minutes? Use α = 0.01.
446
What is normal body temperature?
For many years, doctors have told people that “normal” body
temperature is 98.6 degrees Fahrenheit. Today, we will try to
find out if this is true.
Take your body temperature and record it on whiteboard. Record the following for the data for
the whole class (think of our class as an SRS of all high school students)
sx =
x=
n=
Do the data provide convincing evidence that the mean normal body temperature is different
than the doctor’s claim? Assume the conditions have been met.
Another class did the same activity with these results: x = 97.9
sx = 1.6
n = 30
1. Use T-test on the calculator to find the P-value =
Reject H0 at α = 0.10 ?
Reject H0 at α = 0.05?
Reject H0 at α = 0.01?
2. Use TInterval on the calculator to find the following confidence intervals.
90%:________________
95%:________________
99%:________________
Reject H0?___________
Reject H0?___________
Reject H0?___________
3. What connection do you notice between your answers to #1 and #2?
447
Significance Test for µ
Important ideas:
Check Your Understanding
According to a flyer created by BroadwayPartyRental.com, their 18-inch helium balloons fly,
on average, for 32 hours. You purchase a SRS of 50 18-inch helium balloons from this
company and record how long they fly. You would like to know if the actual mean flight time
of all balloons differs from the advertised 32 hours.
1. State an appropriate pair of hypotheses for a significance test in this setting. Be sure to
define the parameter of interest.
2. A 95% confidence interval for the mean flight time (in hours) for all helium balloons is (28.5,
31.4). Based on this interval, what conclusion would you make for a test of the hypotheses
in #1 at the α = 0.05 significance level?
448
Name: _________________
AP Statistics Handout: Lesson 11.2
Topics: two-sample t-test for a difference of means
Guided Notes
The Stroke Stent Study
In this study*, researchers randomly assigned stroke
patients to two groups: one received the current standard
care (control) and the other received a stent surgery in
addition to the standard care (stent treatment). Mr. YoungSaver’s father, Dr. Jeffrey Saver, was a principal researcher
for the study.
Dr. Saver
Mr. Young-Saver
The paternity is not in question!
Stent in action, working
on a stroke clot
(UT Health San Antonio)
*Saver JL, Goyal M, Bonafe A, Diener HC, Levy EI, Pereira VM, et al. “Stent-retriever thrombectomy after
intravenous t-pa vs. T-pa alone in stroke.” N Engl J Med. 2015; 372:2285-2295
https://www.nejm.org/doi/full/10.1056/nejmoa1415061
Two sample t-test for a difference of means
Study results
Stent
Control
Mean Disability
2.26
3.23
Stdev. Disability
1.78
1.78
98
93
n
If the stents work, the treatment group should
have a ____________________ disability score.
Could the difference in outcome have
happened by chance alone? Or did the
stents actually improve patient outcomes?
449
Hypotheses: Take default belief until evidence convinces us otherwise
Non-numerical version:
Numerical version:
𝐻0 : ___________ standard of care is best
𝐻0 :
• Default/dull belief (___________________)
𝐻𝐴 : Stents ____________ patient outcomes
𝐻𝐴 :
• Unproven research hypothesis we aim to
demonstrate (_______________________)
Where:
𝜇𝑠 is the mean disability score of ____ patients who’d receive stents.
𝜇𝑐 is the mean disability score of ____ patients who’d receive current
standard of care.
The stents don’t improve
outcomes over the current
standard care, so the mean
mRS is the _____________
___________.
The stents did improve care,
so the stent group had a
______________________.
Rewrite these hypotheses in a more mathematically convenient way:
Under the hood: calculating t-test statistics and p-values for a two-sample test
Investigating if we find enough statistical evidence to reject the 𝑯𝟎 and support the 𝑯𝑨
1. Start by assuming the null (the default) is true.
• We assume stents do not help: disability level is same for both groups.
2. Ask yourself: if the null is in fact true, how likely is the
data that you’ve gathered?
2. The study enrolled 191 patients (93 in
control, 98 in treatment). At this sample size,
what estimates for 𝜇𝑠 - 𝜇𝑐 are we ___________
if stents don’t actually help? Let’s find the
______________________________________
of 𝑥̅𝑠 - 𝑥̅𝑐 from all possible random assignments.
Under certain conditions:
~ Norm(𝜇𝑥̅𝑠−𝑥̅𝑐 = 0, 𝜎𝑥̅ 𝑠−𝑥̅𝑐 = √
Under the null assumption, how unlikely was the
data we observed (𝑥̅𝑠 − 𝑥̅𝑐 = −𝟎. 𝟗𝟕)?
450
𝑥̅ ~ 𝑡 (df = 92)
𝜇𝑥̅ = ______
𝑆𝐸𝑥̅ = ______
𝜎𝑠 2
𝑛𝑠
+
𝜎𝑐 2
𝑛𝑐
Write down the t-test statistic and the p-value. Interpret their meaning and describe if they suggest
there is reason to doubt the null hypothesis in this case.
3. Draw a conclusion:
Under my assumption that stents provide no added benefit, the actually observed data (0.97 point
decline in disability in stent group) is highly unlikely (________________). So, I ___________ my earlier
assumption. There’s convincing evidence that the ____________________________________________.
In the stroke stent study, patients were randomly assigned to
receive either the current standard of care or current care with
a stent surgery. The results are summarized below.
Do the results give convincing statistical evidence that the
stent treatment reduces the average disability from stroke?
Mean Disability
Stdev. Disability
n
Stent
2.26
1.78
98
Control
3.23
1.78
93
: Calculator steps for two-sample z-interval for proportions:
For groups 1&2:
𝑥̅ : sample mean
𝑠𝑥 : sample stdev.
n: sample size
µ1: 𝐻𝐴 direction
Pooled → NO
STAT → TESTS →
4: 2-SampTTest…
451
Output
Lesson11.2 Discussion
Researchers enroll patients until reaching a sample size goal. Independent reviewers monitor the trial
data and stop the trial early if either:
1. The treatment seems to be hurting patients
2. The treatment seems so good that it’d be unethical to not give it to everyone.
Discussion Question: You are on the monitoring committee for a new drug that aims to lower
cholesterol among men. Researchers aim to enroll 300 patients in the trial. You look at the results for
the first 150 patients. Would you stop the trial early? Why or why not?
Mean cholesterol
Stdev. cholesterol
n
Control
209 mg/dL
53 mg/dL
75
Treatment
195 mg/dL
55 mg/dL
75
Healthy cholesterol level is < 200 mg/dL
452
11.2 PRACTICE!
The Stronger Picker-upper?
In commercials for Bounty paper towels, the manufacturer claims that they are the “quicker picker-upper.” But
are they also the stronger picker-upper? Two AP Statistics students, decided to find out. They selected a
random sample of 30 Bounty paper towels and a random sample of 30 generic paper towels and measured their
strength when wet. To do this, they uniformly soaked each paper towel with 4 ounces of water, held two
opposite edges of the paper towel, and counted how many quarters each paper towel could hold until ripping,
alternating brands. Here are their results:
Bounty: 106, 111, 106, 120, 103, 112, 115, 125, 116, 120, 126, 125, 116, 117, 114
118, 126, 120, 115, 116, 121, 113, 111, 128, 124, 125, 127, 123, 115, 114
Generic: 77, 103, 89, 79, 88, 86, 100, 90, 81, 84, 84, 96, 87, 79, 90
86, 88, 81, 91, 94, 90, 89, 85, 83, 89, 84, 90, 100, 94, 87
(a) Display these distributions using parallel boxplots and briefly compare these distributions. Based only on the
boxplots, discuss whether or not you think the mean for Bounty is significantly higher than the mean for
generic.
(b) Use a significance test to determine if there is convincing evidence that wet Bounty paper towels can hold
more weight, on average, than wet generic paper towels.
453
Is one form of the AP exam harder?
Last year,South Aiken High School had 30 students take the AP Statistics exam.
They were informed later that the College Board gave two forms of the exam, which
were randomly assigned to the students. Here are the results:
Form A
3
3
3
3
4
4
4
4
5
5
5
5
5
5
5
Form B
2
2
3
3
4
4
4
4
4
5
5
5
5
5
5
Mean score Form A ( x A ) ? ______
Mean score Form B ( x B ) ? ______
What is the difference in means x A − x B ? ___________
Assume the two forms are the same difficulty, so if Doug scored a 5 on Form A, he
would also score a 5 on Form B. In other words, Doug is a 5 no matter which form he is
randomly assigned.
1. The 30 AP scores from the class are written on 30 cards. Randomly assign half
of the students to get Form A and the other half to get Form B. What is the
difference in mean scores for this random assignment?
x A = _________
x B = _________
x A − x B = _________
2. Write the difference of mean scores on a sticker dot and take it to the poster at
the front of the room. Sketch the dotplot below.
difference of mean scores ( x A − x B )
3. East Kentwood had a difference of mean scores of 4.20 – 4.0 = 0.2. Is this
outcome surprising if we assume both forms are the same difficulty? Explain.
4. Based on the simulation, do we have convincing evidence that one form of the
exam is harder? Explain.
454
Significance Test for a Difference in Means
Important ideas:
Check Your Understanding
Last year, South Aiken High School had 30 students take the AP Statistics exam. They
were informed later that the College Board gave two forms of the exam, which were
randomly assigned to the students. The two forms had a difference of mean scores of
4.20 – 4.0 = 0.2. Do the data provide convincing evidence that one form of the AP
exam is harder than the other?
Complete the first two steps of a significance test.
455
Does labeling menus reduce calories?
According to a Stanford Business article, Americans may eat fewer calories at restaurants if
the calories of the food items are labeled on the menu. To investigate this, researchers
compared Starbucks receipts from locations where the menus were labeled to receipts from
stores where the menus were not labeled. A random sample of 30 receipts from stores with
the menus labeled had an average number of calories of 225 calories with a standard
deviation of 100 calories. A random sample of 40 receipts from stores without menus labeled
showed an average of 265 calories per receipt with a standard deviation of 75 calories. Does
this provide convincing evidence that the average calories per receipt at Starbucks with a
labeled menu is less than at a Starbucks without labeled menus?
456
Significance Test for a Difference in Means
Important ideas:
Check Your Understanding
Can balloons hold more air or more water before bursting? A student purchased a large bag of 12-inch
balloons. He randomly selected 10 balloons from the bag and then randomly assigned half of them to be
filled with air until bursting and the other half to be filled with water until bursting. He used devices to
measure the amount of air and water was dispensed until the balloons burst. Here are the data.
Air (ft3)
Water (ft3)
0.52
0.44
0.58
0.41
0.50
0.45
0.55
0.46
0.61
0.38
Do the data give convincing evidence air filled balloons can attain a greater volume than water filled
balloons?
457
Does Memory Training Help? Part 2
Suppose that 32 seniors were randomly selected to try both memory strategies. Each student
calculated the difference in the # words (Strategy 2 – Strategy 1). The mean of the differences was
x diff = 1.5 and the standard deviation of the differences is s diff = 2.7 . Do these data give convincing
evidence that strategy 2 improves memory at the α = 0.05 significance level?
458
Significance Test for A Mean of Differences
Important ideas:
Check Your Understanding
In each of the following settings, decide whether you should use two-sample t procedures to
perform inference about a difference in means or paired t procedures to perform inference about a
mean difference. Explain your choice.
1. A random sample of 30 adults were selected. Each adult reported the number of pieces of
junk mail they received in their mailbox at home that day as well as the number of junk
emails they received in their inbox that day. A researcher would like to know if adults
receive significantly more junk email than junk mail in their mailbox.
2. A random sample of 100 people who live on the west coast and 100 people who live on the
east coast are selected. Each person reports how many spam calls they receive in a oneweek time period. We would like to know if people who live on the west coast receive
significantly more spam calls than those who live on the east coast.
3. A researcher randomly selects a variety of 15 cell phones from different manufacturers and
different service providers. The researcher measures the signal strength from two different
remote areas: one in the state of Pennsylvania and one in the state of Virginia. The
researcher would like to know if the signal strength is significantly weaker in the remote
area in the state of Virginia.
459
Climate Change Part 2
Suppose that 32 weather stations were randomly selected across the globe. For each station, the
peak annual temperature (oC) between 1900-1940 was found. Then, the peak annual temperature
between 1960-2000 was found. Differences were calculated for each station (later temperature –
earlier temperature). The mean of the differences was 𝑥̅!"## = 0.35 oC and the standard
deviation of the differences was 𝑠!"## = 0.296 oC. Do these data give convincing evidence that
the peak global temperatures increased (on average) during the 1900’s?
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
460
Significance Test for A Mean of Differences
Important ideas:
Check Your Understanding
In each of the following settings, decide whether you should use two-sample t procedures to perform
inference about a difference in means or paired t procedures to perform inference about a mean
difference. Explain your choice.
1. A random sample of 30 adults were selected. Each adult reported the number of pieces of
junk mail they received in their mailbox at home that day as well as the number of junk
emails they received in their inbox that day. A researcher would like to know if adults receive
significantly more junk email than junk mail in their mailbox.
2. A random sample of 100 people who live on the west coast and 100 people who live on the
east coast are selected. Each person reports how many spam calls they receive in a oneweek time period. We would like to know if people who live on the west coast receive
significantly more spam calls than those who live on the east coast.
3. A researcher randomly selects a variety of 15 cell phones from different manufacturers and
different service providers. The researcher measures the signal strength from two different
remote areas: one in the state of Pennsylvania and one in the state of Virginia. The
researcher would like to know if the signal strength is significantly weaker in the remote area
in the state of Virginia.
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
461
462
How to find P-value
Formulas for Significance
Tests
Formulas for Confidence
Intervals
Formula for standard
deviation of the sampling
distribution
Formula for mean of the
sampling distribution
Conditions
Name the procedure
Symbol for parameter
(population)
Symbol for statistic
(sample)
Lesson
10.1 – One Mean
AP Stats Chapter 10, 11 Study Sheet
10.2 – Difference of Means
10.2 – Mean Difference
463
2009 AP ® STATISTICS FREE-RESPONSE QUESTIONS (Form B)
5. A bottle-filling machine is set to dispense 12.1 fluid ounces into juice bottles. To ensure that the machine is
filling accurately, every hour a worker randomly selects four bottles filled by the machine during the past hour
and measures the contents. If there is convincing evidence that the mean amount of juice dispensed is different
from 12.1 ounces or if there is convincing evidence that the standard deviation is greater than 0.05 ounce, the
machine is shut down for recalibration. It can be assumed that the amount of juice that is dispensed into bottles
is normally distributed.
During one hour, the mean number of fluid ounces of four randomly selected bottles was 12.05 and the standard
deviation was 0.085 ounce.
(a) Perform a test of significance to determine whether the mean amount of juice dispensed is different from
12.1 fluid ounces. Assume the conditions for inference are met.
(b) To determine whether this sample of four bottles provides convincing evidence that the standard deviation
of the amount of juice dispensed is greater than 0.05 ounce, a simulation study was performed. In the
simulation study, 300 samples, each of size 4, were randomly generated from a normal population with
a mean of 12.1 and a standard deviation of 0.05. The sample standard deviation was computed for each
of the 300 samples. The dotplot below displays the values of the sample standard deviations.
0.01
0.02
0.03
0.04
0.05
0.06
0.07
I. •• • ••• •••••
0.08
0.09
0.10
Use the results of this simulation study to explain why you think the sample provides or does not provide
evidence that the standard deviation of the juice dispensed exceeds 0.05 fluid ounce.
464
Unit 8 & 9
Chapter 12
Inference for
Distributions and
Relationships
465
AP Statistics Handout: Lesson 12.1
Topics: problem of multiple tests, chi-square test for goodness of fit
Guided Notes
Fall, 2014: Harvard is sued by a group of Asian-American
applicants who were rejected in admissions. They claimed racial
discrimination.
Summer, 2018: Courts release reports on Harvard admissions
data, giving an unprecedented window into private University
admissions data. The data we’ll look at today is reconstructed
from parts of the plaintiff report* in which the defense and
plaintiffs generally agreed on the findings.
Today’s Key Analysis: Is there convincing evidence that Harvard discriminates
against Asian applicants?
*Plaintiff report accessible here: https://samv91khoyt2i553a2t1s05i-wpengine.netdnassl.com/wp-content/uploads/2018/06/Doc-415-1-Arcidiacono-Expert-Report.pdf
The problem of multiple tests
Class of 2019 Applicants,
Top 10% Academic Index
Group
%
Class of 2019
Admitted Students
Group
%
Asian-American
57.5%
Asian-American
21.4%
Hispanic
3.1%
Hispanic
12.2%
African-American
0.8%
African-American
11.2%
White
38.7%
White
55.3%
Source: Plaintiff’s report, table B.5.7
Notes:
- Academic Index ratings are internal measures of
academic qualification produced by the Harvard
admissions office. They’re calculated based on
standardized test scores and high school
grades/performance.
- Other groups (Native American, Mixed Race, etc.)
and international students were not included in
the court’s main analysis. Percentages are
calculated just out of these 4 groups.
Source: Harvard Gazette
Imagine Harvard claims: “We only accept the top academic applicants and we treat those applicants
equally. Our admitted class is as good as a random sample from the pool of top applicants.” You’d like to
test if there’s convincing evidence against this claim. Would it be appropriate to do four z-test for
proportions (one for each racial group above)? Why or why not?
Source:
Forbes
466
The null (__________________) hypothesis: there is no
racial discrimination, so the distribution of admitted
students ______________________________________.
Chi-Square test for goodness of fit
Hypotheses
𝐻0 : The racial distribution of admitted students is the ____________ as the claimed distribution
𝐻𝐴 : The racial distribution of admitted students is _____________________ as the claimed distribution
The alternative (______________________) hypothesis:
the distribution of admitted students _______________
________________________________.
Logic of Hypothesis Tests
1. Start by assuming the null (the default) is true. In your own words, describe what that means in
context of the example above:
2. Ask yourself: if the null is in fact true, how likely is the data that you’ve gathered? In your own words,
describe what that means in context of the example above:
3. Draw a conclusion. In your own words, describe what that means in context of the example above:
Calculating the Test Statistic:
Population of Top
Academic Applicants
In total, Harvard admitted n = 2023 applicants from these racial groups to the Class of 2019. How many
would we expect to admit if it was a random sample?
Group
%
Asian-American
57.5%
Hispanic
3.1%
African-American
0.8%
White
38.7%
Expected Counts
Step 1: Assuming
the null is true, this
is what we’d expect
467
Step 2: Compare our null expectation to the actual data
Group
Expected Observed (𝑶𝒃𝒔𝒆𝒓𝒗𝒆𝒅 − 𝑬𝒙𝒑𝒆𝒄𝒕𝒆𝒅)𝟐
Count
Count
𝑬𝒙𝒑𝒆𝒄𝒕𝒆𝒅
AsianAmerican
432
Hispanic
247
AfricanAmerican
226
a) Why do we subtract the observed and
expected values?
b) Why do we square the differences?
c) Why do we divide by the expected values?
White
1118
Calculated with our data:
Formula:
=∑
(𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 − 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑)2
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑
=
When chi-square is _______, that means your observed data is very ________________ from what was
expected under 𝐻0 . Therefore, if 𝐻0 is true, your observations are unlikely (___________________). You
reject 𝐻0 .
When chi-square is __________, that means your observed data is very _____________ to what was
expected under 𝐻0 . Therefore, if 𝐻0 is true, your observations are likely (___________________). You
fail to reject 𝐻0.
Step 3: Conclude → Next page
468
Imagine that Harvard claims: “We only accept the top academic applicants and we treat those applicants
equally. Our admitted class is as good as a random sample from the pool of top applicants.” Do the data
below provide convincing evidence to refute Harvard’s admissions claim?
%
Asian-American
57.5%
Hispanic
3.1%
African-American
0.8%
White
38.7%
Admitted
Top Academic
Applicants
Group
469
Observed
432
247
226
1118
12.1 PRACTICE!
Cheap Dice?
Mrs. Daniel purchased a bunch of inexpensive dice from Amazon. She is now wondering if the dice are fair. So, Mrs.
Daniel randomly selects 6 dice to roll 10 times each, for a total of 60 observations to evaluate. Do we have statistically
convincing evidence that the dice are fair?
Outcome
1
2
3
4
5
6
Total
470
Observed
13
11
6
12
10
8
60
Landline surveys
According to the 2010 Census, of all US residents age 20 and older, 19.1% are in their 20’s, 21.5% are in their 30’s, 21.1
% are in their 40’s, 15.5% are in their 50’s, and 22.8% are 60 and older. The table below shows the age distribution for a
sample of US residents age 20 and older. Members of the sample were chosen by randomly dialing landline telephone
numbers.
Category Count
20-29
141
30-39
186
40-49
224
50-59
211
60+
286
Total
1048
Do these data provide convincing evidence that the age distribution of people who answer landline telephone surveys is
not the same as the age distribution of all US residents?
471
Which color M&M is the most common?
The company that makes milk chocolate M&Ms claims the following distribution:
13% Brown, 14% Yellow, 20% Orange, 16% Green, 24% Blue, and 13% Red. Is this true?
1. Observed values: Brown:_____ Yellow:_____ Orange:_____ Green:_____ Blue:_____ Red:_____
Total number of M&Ms:_______
2. As a class, write down hypotheses for a significance test.
H0:
Ha:
3. Let’s suppose that M&Ms claimed distribution is correct. If they are correct, how many of each
color would we expect to get in our sample.
Expected values: Brown:_____ Yellow:_____ Orange:_____ Green:_____ Blue:_____ Red:_____
Use the table to calculate the test statistic.
Observed
Expected
(Observed - Expected)
(Observed - Expected)2
(Observed - Expected ) 2
Expected
Brown
Yellow
Orange
Green
Blue
Red
Add up all the numbers in the last column. This is our test statistic:_________
4. What value would we get for the test statistic if our sample was very close to what is expected?
Explain.
5. What value would we get for the test statistic if our sample was very far from what is expected?
Explain.
472
Chi-Square Test: Goodness of Fit
Important ideas:
Check Your Understanding
Are births equally likely across the days of the week? A random sample of 150 births give the following
sample distribution:
Day of
the week
Count
Sunday
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
11
27
23
26
21
29
13
a. State the appropriate hypotheses.
b. Calculate the expected count for each of the possible outcomes.
c. Calculate the value of the chi-square test statistic.
d. Which degrees of freedom should you use?
e. Use Table C to find the P-value. What conclusion would you make?
473
Which color M&M is the most common part two?
The company that makes milk chocolate M&Ms claims the following distribution:
13% Brown, 14% Yellow, 20% Orange, 16% Green, 24% Blue, and 13% Red. Is this true?
1. Record the information from yesterday.
Observed values: Brown:_____ Yellow:_____ Orange:_____ Green:_____ Blue:_____ Red:_____
Expected values: Brown:_____ Yellow:_____ Orange:_____ Green:_____ Blue:_____ Red:_____
Test statistic: χ 2 = _________
2. Check conditions:
Random:
10%:
Large counts: Which expected count is the lowest? Are all of the expected counts greater than 5?
3. Calculate the P-value.
For this test df = n – 1, but n represents the number of categories (colors).
What is the df for this test?_______
What is the test statistic for this test?________
Use Table C to find the P-value:________
4. Make a conclusion. Use α = 0.05 .
5. Which color M&M had an observed value the farthest from the expected?
474
Do the data provide significant evidence that the company was lying about the distribution of
colors of M&Ms? Use α = 0.05
475
Chi-Square Test for Goodness of Fit:
Important ideas:
Check Your Understanding
A traffic light is installed to allow traffic from a seldom used side street to cross a 4-lane highway.
Because the side street doesn’t get a lot of traffic the light is set to provide a red light for the side street
80% of the time, yellow 5% of the time, and green 15% of the time. A resident who must pass through
the light several times per day is suspicious that the light is not functioning according to the claimed
distribution. He sets up a trail camera and programs it to snap a picture of the light at 200 randomly
selected times throughout the day. Here are the results: Red: 173, Yellow: 13, and Green: 14.
a. Do these data provide convincing evidence that the light is not functioning according to the claimed
distribution?
b. If there is convincing evidence of a difference in the distribution of car color, perform a follow-up
analysis.
476
Does Harvard Discriminate Against Asian Applicants? (Pt 2)
Below is the true distribution of top academic applicants to Harvard’s Class of 2019, along with the
actually admitted class. Recall: We’re imagining that Harvard claims their admitted students are
essentially a random sample from this pool of top academic applicants. Do the data provide convincing
evidence against Harvard’s claim? Use α = 0.05.
Class of 2019 Applicants with
Top Academic Ratings
Group
%
Class of 2019 Actually
Admitted Students
Group
Count
Asian-American
57.5%
Asian-American
432
Hispanic
3.1%
Hispanic
247
African-American
0.8%
African-American
226
White
38.7%
White
1118
Source: Plaintiff’s report, table B.5.7
Source: Harvard Gazette
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
477
1. Were white applicants admitted at higher or lower rates than expected? Was this surprising? Why or
why not?
3. Discussion Question: Despite the evidence above, how do you believe Harvard’s lawyer would
defend the school’s admissions policy against claims of discrimination? Explain.
4. Discussion Question: What further evidence would you like to know when assessing whether
Harvard’s admissions process is truly discriminatory?
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
478
Chi-Square Test for Goodness of Fit
Important ideas:
Check Your Understanding
A traffic light is installed to allow traffic from a seldom used side street to cross a 4-lane highway. Because
the side street doesn’t get a lot of traffic the light is set to provide a red light for the side street 80% of the
time, yellow 5% of the time, and green 15% of the time. A resident who must pass through the light several
times per day is suspicious that the light is not functioning according to the claimed distribution. He sets up
a trail camera and programs it to snap a picture of the light at 200 randomly selected times throughout the
day. Here are the results: Red: 173, Yellow: 13, and Green: 14.
a. Do these data provide convincing evidence that the light is not functioning according to the claimed
distribution?
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
479
AP Statistics Handout: Lesson 12.2
Topics: chi-square tests for independence and homogeneity
Guided Notes
Force Level
No Force Used
Hands Used
Higher Force Level*
Race of Suspect
Black
Hispanic White
1371
853
260
263
188
30
109
72
18
*Includes push to wall/ground, handcuffs, draw/point weapon, pepper
spray, baton. Data source: NYC.GOV,
https://www1.nyc.gov/site/nypd/stats/reports-analysis/stopfrisk.page
This 2011 data is a random sample of police stops from New York’s “Stop and Frisk” program. The
program allowed police officers to stop people on the street and search them for weapons or
contraband. The program was controversial. Critics alleged that it led to heightened police
discrimination of people of color.
Conditional distributions
i) Fill in the table to the right with the
conditional distribution of force levels among
each racial group:
ii) A television commentator says, “Police had
‘no force’ interactions with only 260 white
suspects. Meanwhile, a much higher number
of black suspects—1,371—didn’t experience
force. Clearly, black people experience ‘forcefree’ interactions with police more frequently
than white people.” Was that a good analysis
of this data? Why or why not?
Black
No Force Used
Hands Used
Higher Force Level
480
White
Imagine a statistician is investigating if there is convincing evidence of an association between race of
suspect and level of force used. The statistician performs a set of z-tests for proportions on the various
rates of force levels used between groups. Why is this not a good idea?
Chi-Square Test for Independence
Hypotheses
𝐻0 : There ____________________________between race of suspects and levels of force used.
• This is the null (dull / default / innocent) hypothesis: force is used at the __________________
between all races.
𝐻𝐴 : There ____________________________ between race of suspects and levels of force used.
• This is the alternative (guilty / proof-needed) hypothesis: force is used at ___________________
between racial groups.
Logic of Hypothesis Tests
1. Start by assuming the null (the default) is true. In your own words, describe what that means in
context of the example above:
2. Ask yourself: if the null is in fact true, how likely is the data that you’ve gathered? In your own words,
describe what that means in context of the example above:
3. Draw a conclusion. In your own words, describe what that means in context of the example above:
481
Calculating the Test Statistic:
Step 1: Assume the null is true. In a world where force is used equally across racial groups, determine
the expected counts of suspects receiving the various levels of force. Show your work in the “Expected
Counts” table below:
Expected Counts
Observed Data
Force
None
Hands
Greater
Total
Black
1,371
263
109
1,743
Expected Count =
Hispanic
853
188
72
1,113
White
260
30
18
308
Total
2,484
481
199
3,164
Force
Black
Hispanic
White
None
2,484
Hands
481
Greater
199
row total × column total
grand total
i) In your own words, explain how the
expected counts were calculated and why
they represent a world under the null
hypothesis assumption:
Total
1,743
1,113
308
Step 2: Compare your expected counts (under null assumption) to the actual observed data
ii) Compare the expected counts to the observed data. What trends do you notice between racial
groups?
Chi-Square Statistic Formula:
=∑
Total
Show the first few terms and final result using our data:
(𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 − 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑)2
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑
=
When chi-square is _______, that means your observed data is very ________________ from what was
expected under 𝐻0. Therefore, if 𝐻0 is true, your observations are unlikely (___________________). You
reject 𝐻0.
When chi-square is _______, that means your observed data is very __________ to what was expected under
𝐻0. Therefore, if 𝐻0 is true, your observations are likely (___________________). You fail to reject 𝐻0.
482
3,164
Step 3: Conclude → We’ll go through that in the process below!
Below are 3,164 randomly sampled records from the 685 thousand stops made by NYC police (as a part
of stop-and-frisk) in 2011. Is there convincing evidence of an association between race of suspect and
level of force used?
Race of Suspect
Force Level
Black
Hispanic White
No Force Used
1371
853
260
Hands Used
263
188
30
Higher Force Level*
109
72
18
*Includes push to wall/ground, handcuffs, draw/point weapon, pepper
spray, baton. Data source: NYC.GOV,
https://www1.nyc.gov/site/nypd/stats/reports-analysis/stopfrisk.page
Calculator steps
Data
2nd → MATRIX→
Edit → [A]
Input Observed
STAT → TESTS →
C: 𝑿𝟐 − 𝐓𝐞𝐬𝐭 …
Observed: [A]
Expected: [B]
Expected Values
483
Output
2nd → MATRIX→
Edit → [B]
Chi-Square Test for Homogeneity
It’s the _______________________ as the chi-square test for independence. With two key differences:
• The data ________________
• The ________________________and (therefore) conclusions
Independence Situation
Homogeneity Situation
Data “setup” - We are testing the association
of…
• ____________________: race of suspect
and force level
• One sample from __________________:
stops in NYC
Data “setup” - We are testing the association
of…
• ___________________: race of suspects
• 3 samples from ___________________:
NYC, Boston, and LA stops
City
Force Level
No Force Used
Hands Used
Higher Force Level
Race of Suspect
Black
Hispanic White
1371
853
260
263
188
30
109
72
18
Race of Suspect
Black
Hispanic
N.Y.C Boston
1,743 457
1,113 1,034
L.A.
1,367
1,467
White
Asian
Native American
308
126
56
780
245
145
893
76
22
Note: this is fake data
i) Write the hypotheses for this situation:
i) Write the hypotheses for this situation:
ii) In your own words, what makes this an
independence situation, rather than a homogeneity
situation?
ii) In your own words, what makes this a
homogeneity situation, rather than an independence
situation?
484
Lesson 12.2 Discussion
Force level
Black
Hispanic
White
No Force
78.7%
76.6%
84.4%
Hands Used
15.1%
16.9%
9.7%
Higher Force
6.3%
6.5%
5.8%
Recap: We found that race of suspect and level of
force used were significantly associated. In other
words, the differences in use of force rates between
racial groups were unlikely due to chance alone.
Discussion Questions
i) Are the differences in force rates across racial groups practically important?
ii) Does our data provide enough evidence to prove that New York police are racially biased (in terms of
use of force)? Why or why not?
485
12.2 PRACTICE!
Ibuprofen or Acetaminophen?
In a study reported by the Annals of Emergency Medicine, researchers conducted a randomized, double-blind clinical
trial to compare the effects of ibuprofen and acetaminophen plus codeine as a pain reliever for children recovering from
arm fractures. There were many response variables recorded, including the presence of any adverse effect, such as
nausea, dizziness, and drowsiness. Is there evidence to suggest a difference between the drug and the side effects?
Here are the results:
Ibuprofen
Adverse effects
No adverse effects
Total
36
86
122
486
Acetaminophen
Total
plus Codeine
57
93
55
141
112
234
Tide vs New Tide
Before bringing a new product to market, firms carry out extensive studies to learn how consumers react to the product
and how best to advertise its advantages. Here are data from a study of a new laundry detergent. The participants are a
random sample of people who don’t currently use the established brand that the new product will compete with. Give
subjects free samples of both detergents. After they have tried both for a while, ask which they prefer. The answers may
depend on other facts about how people do laundry.
Determine whether or not the sample provides convincing evidence that laundry practices and product preference are
independent in the population of interest.
487
Does gummy bear brand matter?
Is the distribution of gummy bear color the same for Haribo gummy bears and Meijer
gummy bears? We’ll collect data as a class and determine if we have convincing evidence
of a difference.
1. Add your data to the board and fill in the table below with the class totals.
Observed:
Brand
Haribo
Meijer
Total
Red
Green
Color
Yellow
Orange
White
Total
2. How many samples do we have? What population are they from? Explain.
3. How many variables are we examining? Explain.
4. As a class, write down hypotheses for a significance test.
H0:
Ha:
5. Now we will use a chi-square test to test if there is a difference between the two
populations. We first need to find the expected values. Complete the table below.
Expected:
Brand
Haribo
Meijer
Red
Green
Color
Yellow
Orange
White
Total
488
Total
6. Use your work on the front page to complete a significance test.
df = (rows – 1)(columns – 1)
7. Explain how this test is different from a chi-square test for goodness of fit?
489
Chi-Square Test for Homogeneity
Important ideas:
Check Your Understanding
A social scientist selects a random sample of 25 freshmen, 25 sophomores, 25 juniors, and 25 seniors
from various high schools across the state Kentucky. Each student was asked if they preferred inperson or remote learning. Here are the results:
Remote
In Person
Freshman
3
22
Sophomore
12
13
Junior
14
11
Senior
15
10
a. State the appropriate null and alternative hypotheses.
b. Show the calculation for the expected count in the Remote/Senior cell. Then provide a
complete table of expected counts.
c. Calculate the value of the chi-square test statistic.
490
Are Taco Tongue and Evil Eyebrow independent?
Is there an association between the Taco Tongue and the Evil Eyebrow? Below is the data
for a random sample of 600 Senior students. Do we have convincing evidence that the
ability to do the Taco Tongue and Evil Eyebrow are associated for all Seniors?
1. Describe what it means for two events to be independent.
2. Calculate the expected counts.
Observed:
Expected:
3. Do the data provide significant evidence that there is an association between the ability to Taco
Tongue and Evil Eyebrow for all Seniors? Use α = 0.05
491
Chi-Square Test for Independence
Important ideas:
Check Your Understanding
For each of the following situations decide what type of chi square test is appropriate. Explain.
1. A random sample of 200 students was asked to sample a new type of pizza that the school was
considering using as a replacement for the current pizza. Each student stated if they were a
freshman, sophomore, junior, or senior and also if they liked the new pizza more than the current
pizza (or not). The school would like to know if there is a relationship between grade level and
pizza opinion.
2. Another school is also considering changing their pizza vendor. This school selects separate
random samples of 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors. Each student tries
the new pizza and states whether they like it more than the current pizza (or not). The school
would like to know if the distribution of opinion differs across the grade levels.
3. A pizza shop claims that 30% of orders are placed on Fridays, 20% are placed on Saturdays,
and 10% of orders are placed on the other days of the week. A global pandemic may have
changed this distribution. The manager investigates so he knows how to staff the pizza shop
appropriately. He selects a random sample of 300 orders and classifies each one according to
the day of the week the order was placed. He wants to know if the distribution of orders is the
same as it was before the global pandemic.
492
Was “Stop and Frisk” Biased?
Today, we’ll analyze 2011 data from New York’s “Stop and Frisk” program. The program allowed
police officers to stop people on the street and search them for weapons or contraband. The
program was controversial. Critics alleged that it led to heightened police discrimination of people of
color. We will explore that claim using a chi-square test for independence.
Open the following links:
• The program data (may take some time to load): tinyurl.com/stop-and-frisk-data
• Random number generator: random.org/integers
• NYPD precinct map: tinyurl.com/nypd-precincts
The data we’re using contains every stop made by NYPD police officers in 2011 (more than half a
million stops in total). Each row represents a single stop. The second tab in the spreadsheet
contains a data key and further information. Note: These data were reported by the police officers
who made the stops.
1. Use the random number generator to obtain 10 random integers between the values 2 and
632722 (the number of dataset rows). Find the dataset row numbers that correspond with your 10
random integers and record the following information for each selected stop:
Race
Force
Level
2. For one of your selected stops, look at all the variables listed: race, gender, and age of suspect
along with the suspected crime and whether an arrest was actually made. If you’d like, use the
precinct number and precinct map (linked above) to see the general area in which this stop took
place. Write a two-sentence description of this stop, as if you’re writing a news brief for an article:
3. Often, police interactions are portrayed at an individual level in the news media, with vivid details
given about the people involved and the interactions themselves. We are about to conduct a
statistical analysis of many interactions, focusing solely on the relationship between race and force
level used. What are the strengths and weaknesses of each type of analysis (individual &
statistical)?
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
493
4. As a class, combine your samples to fill in the observed table. Then, calculate the expected
counts. Note: all force levels other than “hands” (including push to wall/ground, handcuffs,
draw/point weapon, pepper spray, baton) are all considered “higher level” force. The NYPD did not
include police shootings in this dataset.
Observed
Force
Black
Hispanic
Expected
White
Total
Force
None
None
Hands
Only
Hands
Only
Higher
Level
Higher
Level
Total
Total
Black
Hispanic
White
Total
5. Do the data provide convincing evidence of an association between race of suspects and the
levels of force used by police officers? Use α = 0.05.
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
494
The population data (all 2011 stops):
Raw Counts
Force Level
Black
None
Hispanic
274,260
Conditional Distribution (By Race)
White
Total
170,633 52,040 496,933
Hands Only
52,676
37,620
5,904
96,200
Higher Level
21,739
14,321
3,528
39,588
Total
348,675
222,574 61,472 632,721
Force level
Black
Hispanic
White
None
78.7%
76.7%
84.7%
Hands Only
15.1%
16.9%
9.6%
Higher Level
6.2%
6.4%
5.7%
There is a substantial association between race of suspects and the force levels used in the population.
Specifically, White suspects receive “no force” at a higher rate, and they receive higher level force at a
lower rate.
Discussion Questions
1. Were the results of your chi-square test consistent with the above conclusion about the population
data? Explain.
2. Do these data provide enough evidence to prove that New York police are racially biased (in terms of
use of force)? Why or why not?
3. (Optional Extension Question) Information about the residential population of NYC in 2011 is
displayed below. Compare these counts to the data above. If police stops were done completely at
random, which groups would be stopped proportionally more often? Less often? What conclusions can
you draw from this?
Race
Black
Hispanic
White
Population
2,054,101
2,373,304
2,731,173
Data source: 2011 American Community Survey (data.census.gov)
Lesson provided by Stats Medic (statsmedic.com) & Skew The Script (skewthescript.org)
Made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
495
Coincidence or Murder?
Kristen Gilbert started working at the VAMC as a nurse in 1989. A proficiency report
obtained by the Boston Globe described her as “highly skillful,” calm and
compassionate. She organized charity drives and collections for the needy, one of her
defense attorneys said, and she once organized a memorial service for a colleague who
died of cancer.
However, Gilbert did have a tendency to be nearby when patients died. Co-workers
sometimes referred to her as the “angel of death,” the Boston Globe reported.
Federal prosecutor Bill Welch said that death rates seemed to increase significantly
during the shifts Gilbert was assigned to, according to the Globe. Eventually, her fellow
nurses grew suspicious.
In particular, Gilbert’s co-workers started to raise concerns about an increase in medical
emergencies (“codes”) and deaths in Ward C between August 1995 and February 1996,
court records show. A criminal investigation was launched in February, and Gilbert left
her job soon after learning she was the target of it.
According to a search warrant affidavit, a statistical analysis of codes and deaths in
Ward C between January 1, 1995 and February 19, 1996 showed that Gilbert was
present or on duty for 37 of them. Authorities determined the likelihood that her frequent
presence during these emergencies was merely coincidental was _________.
1. What other explanations might there be for why Gilbert had more deaths on her
shifts?
2. What statistical information would we need to know in order to make a decision
about whether the deaths on Gilbert’s shifts were coincidental, or in other words,
purely by chance?
496
An analysis of 1641 eight-hour shifts is presented below. Use the data to decide if there
is an association between Gilbert’s presence and death during a shift.
Death during shift?
Gilbert present?
Yes
No
Yes
40
217
No
34
1350
3. What type of test will you use to analyze the data? Why?
4. Conduct the test below using the 4 step process.
5. Do you think Kristen Gilbert murdered her patients? Explain using the results of
your study.
497
AP Statistics Handout: Lesson 12.3
Topics: chi-square vs. regression, t-interval for regression slope, t-test for regression slope
Note: Lesson inspired by “Silence the
Violence” materials from 21st Century
Math Projects
Guided Notes
Which country is this?
Data notes: Random sample of developed
nations from University of Sydney School of
Public Health database. Full database available
at gunpolicy.org. Data from most recent year
available for each nation (2014-2019).
"Developed" nations = top 65 countries in
2019 UN Human Development Index.
Houston Gun Show, photo by glasgows
Cartoon by David Simonds
Two positions for preventing gun homicide:
Gun Control
More Guns
Describe the arguments both of these sides make, as well as the nations they reference when making
these arguments:
Source:
Forbes
498
Chi-Square vs. Regression
Nation
Australia
Belgium
Bulgaria
Czech Republic
Finland
France
Germany
Greece
Hungary
Italy
Japan
Lithuania
Malta
Netherlands
Portugal
Qatar
Romania
Switzerland
Sweden
Civilian
Firearms per
100 People
Gun Homicides
per 1,000,000
People
13.7
12.7
8.4
12.5
32.4
19.6
32.0
17.6
10.55
12.89
0.3
13.6
28.3
2.6
24.22
16.7
2.6
34.8
23.1
1.5
2.5
2.0
1.0
2.0
1.2
0.6
1.9
0.5
2.9
0.1
1.1
4.6
1.6
2.4
0.5
0.3
1.6
4.0
Random sample of 19 developed nations
i) How many variable(s) are there in this dataset?
ii) Are the variable(s) quantitative or categorical?
iii) When we have two categorical variables, how do we
organize and test the data? What about two quantitative
variables?
iv) In this case, which variable is the explanatory and which
is the response? How do you know?
Data notes: Random sample of developed nations from University
of Sydney School of Public Health database. Full database available
at gunpolicy.org. Data from most recent year available for each
nation (2014-2019). "Developed" nations = top 65 in 2019 UNHDI.
Scatterplot and Least Squares Regression Line (LSRL)
i) Circle and label Japan and Germany on these graphs. Comment on how their locations in these graphs
relate to the “gun control” vs. “more guns” arguments:
ii) Comment on the overall direction and strength of the relationship:
499
t-Interval for the Slope of a Regression Line
Predictor
Constant
Firearms
S = 1.141
Coef
SE Coef
0.91618 0.51205
0.04675 0.02625
R-Sq = 15.73%
T
1.789
1.781
P
0.0914
0.0928
R-Sq (adj) = 10.77%
1) Write down the LSRL equation:
2) Interpret the estimate for the slope:
Making the Interval
Construct and interpret a 95% confidence interval for the population regression line. First, identify the
following:
Population:
Parameter:
Sample:
Statistic:
β
95% confident we’ll capture this
Statistic ± margin of error
Statistic ± (critical value)(standard error)
_________________ standard errors in
either direction.
Using the formula, the computer output, and technology,
calculate and interpret the t-interval for the slope of the
regression line:
Confidence Interval for Regression Slope:
𝑏 ± (𝑡 ∗ )(𝑆𝐸𝑏 )
To find 𝒕∗ with TI-84:
2nd → VARS → 4: InvT
500
-Area: percent below
interval (2.5% = 0.025)
-df: n – 2 (19 – 2 = 17)
t-Test for the Slope of a Regression Line
Predictor
Constant
Coef
0.91618
SE Coef
0.51205
T
1.789
P
0.0914
Firearms
0.04675 0.02625
1.781
0.0928
S = 1.141
R-Sq = 15.73%
R-Sq (adj) = 10.77%
Is there convincing evidence of a linear relationship between rates of gun ownership and rates of gun
homicide in developed nations?
Hypotheses:
𝐻0 : 𝛽 = 0
𝐻𝐴 : 𝛽 ≠ 0
β: The true slope of the population regression
line relating rates of gun ownership (per 100
people) and rates of gun homicide (per 1
million people) in developed nations.
i) In your own words, interpret what the null and alternative hypotheses mean in this context:
ii) Why do our hypotheses use β instead of b?
iii) Why do you think the above test is the same thing as testing if there is no correlation (𝜌 = 0)
between rates of gun ownership and homicide?
Conducting the test:
iv) You can use the interval we calculated earlier to figure out the result of this significance test. Please
do so and explain your results:
v) Report the p-value and test statistic. Circle where they came from in the computer output above.
Then, write the conclusion of your test:
501
Rates of gun ownership and rates of gun homicide from 19 randomly sampled developed nations is
shown in the scatterplot below. LSRL computer output, a residual plot, and a histogram of the residuals
are provided:
Predictor
Constant
Firearms
S = 1.141
Coef
SE Coef
0.91618 0.51205
0.04675 0.02625
R-Sq = 15.73%
T
1.789
1.781
P
0.0914
0.0928
R-Sq (adj) = 10.77%
Conditions for Regression Inference
L - _______________, to check this condition:
I - ________________, to check this condition:
N - _______________, to check this condition:
E - _______________, to check this condition:
R - _______________, to check this condition:
Plots A & B are made from mock data. If you were performing inference on their datasets, these plots
would be a bad sign for your conditions. Which conditions would they violate and why?
A
B
502
Lesson 12.3 Discussion
We ______________________ convincing evidence of a relationship between rates of gun ownership
and rates of gun homicide. This conclusion makes sense: the relationship was ___________________.
Original Sample (USA not included)
With USA Added
Discussion Question: Would reducing civilian gun ownership also reduce gun homicides in the U.S.?
Explain.
•
•
•
First: Answer this question using only the data we’ve analyzed, without making any assumptions.
Second: Answer by choosing a clear side and use a BS (Bad Statistics) argument to support it.
Third: Provide your actual personal opinion. Make sure, as always, to address the data.
503
Does seat location matter? Part 1
Do students who sit in the front rows do better than students who sit farther away?
Row
1
1
Score 76 77
1
94
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
99 88 90 83 85 74 79 77 79 90 88 68 78 83 79
Row
4
4
4
4
4
4
5
5
5
5
5
5
Score 94 72 101 70 63 76 76 65 67 96 79 96
1. Is this an observational study or an experiment? Why?
2. Why is it important to randomly assign the students to seats rather than letting each student choose
his or her own seat?
3. How many variables are we measuring?_____ Are they categorical or quantitative?_____________
What is the explanatory variable (x)?__________________ Response variable(y)?_____________
4. Use stapplet.com to make a scatterplot. Sketch it below.
5. Find the least squares regression line (LSRL):__________________________________________
6. What is the slope of the LSRL:_________
Interpret the slope in the context of the problem.
504
Does the negative slope provide convincing evidence that sitting closer causes higher achievement, or is
it plausible that the association is purely by chance because of random assignment?
In order to answer this question, we need to know more about “purely by chance because of random
assignment”. If we assume that seat location has NO effect on Exam Score, then we could just randomly
assign all 30 Exam Scores to each of the seat locations. We will do this by writing down each of the 30
Exam Scores onto an index card, shuffle the index cards, and then randomly assign them to seat
locations.
In pairs, shuffle up the note cards and randomly assign 6 students into each of the 5 rows. Record the
results:
Row 1: _____, _____, _____, _____, _____, _____
Row 2: _____, _____, _____, _____, _____, _____
Row 3: _____, _____, _____, _____, _____, _____
Row 4: _____, _____, _____, _____, _____, _____
Row 5: _____, _____, _____, _____, _____, _____ Now find the slope of the LSRL:____________
Repeat this process 2 more times for a total of 3 different samples. Record the results.
Row 1: _____, _____, _____, _____, _____, _____
Row 2: _____, _____, _____, _____, _____, _____
Row 3: _____, _____, _____, _____, _____, _____
Row 4: _____, _____, _____, _____, _____, _____
Row 5: _____, _____, _____, _____, _____, _____ Now find the slope of the LSRL:____________
Row 1: _____, _____, _____, _____, _____, _____
Row 2: _____, _____, _____, _____, _____, _____
Row 3: _____, _____, _____, _____, _____, _____
Row 4: _____, _____, _____, _____, _____, _____
Row 5: _____, _____, _____, _____, _____, _____ Now find the slope of the LSRL:____________
You have now calculated three different possible values for the slope based on random assignment.
Take these 3 values to the dotplot on the whiteboard in the front of the room. When everyone in class
has recorded their data, copy the dotplot below:
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Sampling Distribution of b
Important ideas:
Check Your Understanding
You may have heard that your nose and ears grow through your whole life. While it is true that your
nose and ears get bigger throughout life, its not because they grow, but because of gravity. The
cartilage in your nose and ears break down as we age and the “growth” people observe is the result
of drooping. To quantify the expansion of ears over time, a random sample of 30 adults were
selected. For each adult, their age (in years) was recorded and their ear height (cm) was measured.
Below is the regression output. Is there evidence of a positive linear relationship between age and
ear height? Assume the conditions for inference are met.
a. What is the estimate for α? Interpret this value.
b. What is the estimate for β? Interpret this value.
c. What is the estimate for σ? Interpret this value.
d. Give the standard error of the slope SEb. Interpret this value.
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Does seat location matter – Part 2?
Do students who sit in the front rows do better than students who sit farther away? Mrs.
Gallas took a random sample of 30 students from her classes and found these results.
Row
1
1
Score 76 77
1
94
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
99 88 90 83 85 74 79 77 79 90 88 68 78 83 79
Row
4
4
4
4
4
4
5
5
5
5
5
5
Score 94 72 101 70 63 76 76 65 67 96 79 96
Line of best fit:________________
Slope: b =
SEb = 1.33
1. If Mrs. Gallas were to take another random assignment of 30 students, do you think the slope of the
LSRL would be the same? Why?
2. We are going to construct a 95% confidence interval for the slope of the population regression line.
Identify the parameter and statistic.
Parameter:________________________________
3. There are five conditions to check.
(1) Linear: The scatterplot needs to show a linear
relationship. Also, the residual plot doesn’t have a
leftover curved pattern. Sketch each at right.
(2) Independent:
(3) Normal: A dotplot of the residuals cannot show
strong skew or outliers. Make one using the applet
and sketch it at right.
(4) Equal SD: The residual plot does not show a clear
sideways Christmas tree pattern.
(5) Random:
4. Construct the interval:
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Statistic:____________
Confidence Intervals for Slope
Important ideas:
Check Your Understanding
A thrill-seeker wanted to try to travel across a large field while being suspended in the air by holding
onto balloons. In order to determine the number of balloons needed per pound of weight, he did a
preliminary study. He selects a random sample of 20 rocks of various sizes. He weighed each one
and also determined how many balloons are needed to lift the rock. Here is output from a leastsquares regression analysis of the data.
Construct and interpret a 90% confidence interval for the slope of the population regression line.
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How does GPA relate to ACT score?
Mrs. Gallas is wondering if there is a relationship between GPA and ACT score. She took a
random sample of 9 out of her 101 students and recorded their GPA and ACT score. The
data are below.
Student #
GPA
ACT
83
3.7
23
69
2.3
20
96
4.0
35
89
3.8
33
57
3.0
22
13
1.8
13
24
2.0
17
37
2.3
20
1. What relationship would you expect GPA and ACT score to have? Explain.
Here is the Minitab ouput as well as graphs of the data.
Predictor
Coef
SE Coef
T
P
Constant
1.201
0.0874
13.72
0
GPA
7.507
1.29
5.82
0.0006511
S = 3.252686
R-Sq = 82.8%
R-Sq(adj) = 76.5%
2. Find the LSRL for the data.
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91
3.9
29
3. Do the data provide significant evidence that there is a positive linear relationship
between GPA and ACT?
510
Significance Test for Slope
Important ideas:
Check Your Understanding
You may have heard that your nose and ears grow through your whole life. While it is true that your
nose and ears get bigger throughout life, it’s not because they grow, but because of gravity. The
cartilage in your nose and ears break down as we age and the “growth” people observe is the result
of drooping. To quantify the expansion of ears over time, a random sample of 30 adults were
selected. For each adult, their age (in years) was recorded and their ear height (cm) was measured.
Below is the regression output. Is there convincing evidence of a positive linear relationship
between age and ear height? Assume the conditions for inference are met.
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512
513
2011 AP ® STATISTICS FREE-RESPONSE QUESTIONS
5. Windmills generate electricity by transferring energy from wind to a turbine. A study was conducted to examine
the relationship between wind velocity in miles per hour (mph) and electricity production in amperes for one
particular windmill. For the windmill, measurements were taken on twenty-five randomly selected days, and the
computer output for the regression analysis for predicting electricity production based on wind velocity is given
below. The regression model assumptions were checked and determined to be reasonable over the interval of
wind speeds represented in the data, which were from 10 miles per hour to 40 miles per hour.
Predictor
Constant
Wind velocity
Coef
0.137
0.240
S = 0.237
R-Sq = 0.873
SE Coef
0.126
0.019
T
1.09
12.63
p
0.289
0.000
R-Sq (adj) = 0.868
(a) Use the computer output above to determine the equation of the least squares regression line. Identify all
variables used in the equation.
(b) How much more electricity would the windmill be expected to produce on a day when the wind velocity is
25 mph than on a day when the wind velocity is 15 mph? Show how you arrived at your answer.
(c) What proportion of the variation in electricity production is explained by its linear relationship with wind
velocity?
(d) Is there statistically convincing evidence that electricity production by the windmill is related to wind
velocity? Explain.
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EXAM REVIEW
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AP Statistics Chapter 1 Notes - Exploring Data
1.1/2: Categorical Variables and Displaying Distributions with Graphs
Individuals and Variables
 Individuals are objects described by a set of data. Individuals may be people, but they
may also be animals or things.
 A variable is any characteristic of an individual. A variable can take different values for
different individuals.
Categorical and Quantitative Variables
 A categorical variable places an individual into one of several groups or categories.
 A quantitative variable takes numerical values for which arithmetic operations such as
adding and averaging make sense.
Distribution
The distribution of a variable tells us what values the variable takes and how often it takes
these variables.
Describing the Overall Pattern of a Distribution – Remember your SOCS
To describe the overall pattern of a distribution, address all of the following:
 Spread – give the lowest and highest value in the data set
 Outliers – are there any values that stand out as unusual?
 Center – what is the approximate average value of the data (only an estimation)
 Shape – does the graph show symmetry, or is it skewed in one direction (see below)
Outliers
An outlier in any graph of data is an individual observation that falls outside the overall
pattern of the graph.
Describing the SHAPE of a distribution – Symmetric and Skewed Distributions
Symmetric
Skewed Left
Skewed Right
Mean = Median
Mean < Median
Mean > Median
Time Plot
 A time plot of a variable plots each observation against the time at which it was
measured.
 Always mark the time scale on the horizontal axis and the variable of interest on the
vertical axis. If there are not too many points, connecting the points by lines helps show
the pattern of changes over time.
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1.3: Describing Distributions with Numbers
The Mean ( x )
To find the mean of a set of observations, add their values and divide by the number of
observations. If the n observations are x1, x2, …, xn, their mean is:
x  x  x    xn
x 1 2 3
n
n
or simply,
x   xi
i 1
The Median (M)
 The median M is the midpoint of distribution, the number such that half the observations
are smaller and the other half are larger. To find the median of distribution:
 Arrange all observation in order of size, from smallest to largest.
 If the number of observations n is odd, the median M is the center observation in the
ordered list. The position of the center observation can be found at (n + 1) / 2
 If the number of observations n is even, the median M is the mean of the two center
observations in the ordered list. The position of the two middle values are n/2 and n/2 + 1
The Five-Number Summary
The five-number summary of a data set consists of the smallest observation, the first quartile,
the median, the third quartile, and the largest observation, written in order from smallest to
largest. In symbols, the five-number summary is:
Minimum – Q1 – M – Q3 – Maximum
The Quartiles (Q1 and Q3 )
 To calculate the quartiles, arrange the observations in increasing order and locate the
median M in the ordered list of observations.
 The 1st quartile (Q1) is middle number of the values that are less than the median.
 The 3rd quartile (Q3) is the middle number of the values that are greater than the median.
Example
2
14
28
29
30
32
33
34
40
42
52
Min
Q1
Med
Q3
Max
The Interquartile Range (IQR)
The IQR is the distance between the first and third quartiles, IQR = Q3 - Q1
Outliers: The 1.5 x IQR Criterion
Call an observation an outlier if it falls more than 1.5 x IQR below the first quartile or above
the third quartile. Using the 5-number summary from above as an example (IQR = 40-28=12)
 Low outlier cutoff: Q1  1.5  IQR (example: 28 – 1.5(12) = 28 – 18 = 10) Therefore,
the 2 is an outlier.
 High outlier cutoff: Q3  1.5  IQR (example: 40 + 1.5(12) = 40 + 18 = 58) no outlier
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1.3: Describing Distributions with Numbers
Boxplot
A boxplot is a graph of the five-number summary, with outliers plotted individually.
 A central box spans the quartiles.
 A line in the box marks the median.
 Observations more than 1.5 x IQR outside the central box are plotted individually.
 Lines extend from the box out to the smallest and largest observations, not the outliers.
Example:
The Standard Deviation (S or Sx)
The standard deviation of a set of observations is the average of the squares of the deviations
of the observations from their mean. The formula for the standard deviation of n observations
x1, x2, …, xn is:
s
 ( x  x)
2
i
n 1
Calculation of the Standard Deviation
Consider the data below which has a mean of 4.8:
xi
6
3
8
5
2
Sum
xi – mean
6 – 4.8 = 1.2
3 – 4.8 = -1.8
8 – 4.8 = 3.2
5 – 4.8 = 0.2
2 – 4.8 = -2.8
0
So the standard deviation is
(xi-mean)2
(1.2)2 = 1.44
(-1.8)2 = 3.24
(3.2)2 = 10.24
(0.2)2 = 0.04
(-2.8)2 = 7.84
22.8
22.8 / (5  1)  22.8 / 4  5.7  2.387
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AP Statistics Chapter 2 – Describing Location in a Distribution
2.1: Measures of Relative Standing and Density Curves
Density Curve
A density curve is a curve that
• is always on or above the horizontal axis, and
• has area exactly 1 underneath it.
A density curve describes the overall pattern of a distribution. The area under the curve and
above any range of values is the proportion of all observations that fall in the range.
Example
The density curve below left is a rectangle. The area underneath the curve is 4 i 0.25 = 1.
The figure on the right represents the proportion of data between 2 and 3 ( 1 i 0.25 = 0.25 ).
Median and Mean of a Density Curve
• The median of a density curve is the equal-areas point, the point that divides the area
under the curve in half.
• The mean of a density curve is the balance point, at which the curve would balance if
made of solid material.
• The median and mean are the same for a symmetric density curve. They both lie at the
center of the curve. The mean of a skewed curve is pulled away from the median in the
direction of the long tail.
Normal Distributions
A normal distribution is a curve that is
• mound-shaped and symmetric
• based on a continuous variable
• adheres to the 68-95-99.7 Rule
The 68-95-99.7 Rule
In the normal distribution with mean μ and standard deviation σ:
• 68% of the observations fall within 1σ of the mean μ.
• 95% of the observations fall within 2σ of the mean μ.
• 99.7% of the observations fall within 3σ of the mean μ.
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2.2: Normal Distributions
Standardizing and z-Scores
If x is an observation from a distribution that has mean μ and standard deviation σ, the
standardized value of x is
z=
x−μ
σ
A standardized value is often called a z-score.
Standard Normal Distribution
• The standard normal distribution is the normal distribution N(0, 1) with mean 0 and
standard deviation 1.
• If a variable x has any normal distribution N(μ, σ) with mean μ and standard deviation σ,
then the standardized variable
x−μ
z=
σ
has the standard normal distribution (see diagram below).
The Standard Normal Table
Table A is a table of areas under the standard normal curve. The table entry for each value z is
the area under the curve to the left of z.
Standard Normal Calculations
Area to the left of z ( Z < z ) Area to the right of z ( Z > z )
Area = Table Entry
Area = 1 – Table Entry
Area between z1 and z2
Area = difference between
Table Entries for z1 and z2
Inverse Normal Calculations
Working backwards from the area, we find z, then x. The value of z is found using Table A in
reverse. The value of x is found, from z, using the formula below
x = μ + z iσ
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AP Statistics Chapter 3 – Examining Relationships
3.1: Scatterplots and Correlation
Explanatory and Response Variables
A response variable measures an outcome of a study. An explanatory variable attempts to
explain the observed outcomes. The explanatory variable is sometimes referred to as the
independent variable and is typically symbolized by the variable x. The response variable is
sometimes referred to as the dependent variable and is typically symbolized by the variable y.
Scatterplot
A scatterplot shows the relationship between two quantitative variables measured on the same
individuals. The values of the explanatory variable appear on the horizontal axis, and the values
of the response variable appear on the vertical axis. If there is no clear explanatory/response
relationship between the two variables, then either variable can be placed on either axis. Each
individual in the data set appears as a single point in the plot fixed by the values of both variables
for that individual.
Examining a Scatterplot
In any graph of data, look for patterns and deviations from the pattern. Describe the overall
pattern of a scatterplot by the form, direction and strength of the relationship.
 Form can be described as linear or curved.
 Direction can be described as positive or negative or neither.
 Strength can be described as weak, moderate or strong.
A deviation from the overall pattern of a scatterplot is called an outlier.
Association
 Two variables are positively associated if as one increases the other increases.
 Two variables are negatively associated if as one increases the other decreases.
Correlation
Correlation measures the strength and direction of the relationship between two quantitative
variables. Correlation is usually represented by the letter r.
Facts about Correlation
1. When calculating correlation, it makes no difference which variable is x and which is y.
2. Correlation is only calculated for quantitative variables, not categorical.
3. The value of r does not change if the units of x and/or y are changed.
4. Positive r indicates a positive association between x and y. Negative r indicates a
negative association.
5. Correlation is always a number between -1 and +1. Values close to +1 or -1 indicate that
the points lie close to a line. The extreme values of +1 and -1 are only achieved when the
points are perfectly linear.
6. Correlation measures the strength of a linear relationship between two variables, not
curved relationships.
7. Correlation, like the mean and standard deviation, is nonresistant. Recall that this means
that it is greatly affected by outliers.
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3.2: Least-Squares Regression
Regression Line
A regression line is a straight line that describes how a response variable y changes as an
explanatory variable x changes. The line is often to predict values of y for given values of x.
Regression, unlike correlation, requires an explanatory/response relationship. In other words,
when x and y are reversed, the regression line changes. Recall that correlation is the same no
matter which variable is x and which is y.
Least-Squares Regression Line
The least-squares regression line is the line that makes the sum of the squares of the vertical
distances from the data points to the line as small as possible.
Equation of the Least-Squares Regression Line
To find the equation of the regression line in the form y  a  bx , where a is the y-intercept and b
is the slope, use the following equations:
br
sy
sx
and a  y  bx
The Role of r-squared (Coefficient of Determination)
The square of the correlation coefficient, or r-squared, represents the percentage of the change
in the y-variable that can be attributed to its relationship with the x-variable. So if r-squared for
the regression between x and y is .73, we can say that x accounts for 73% of the variation in y.
Residuals
A residual is the difference between an observed value of y and the value predicted by the
regression line. That is, residual = actual y - predicted y.
Residual Plot
A residual plot is a scatterplot of each x-value and its residual value. The residual plot is used to
determine whether a linear equation is a good model for a set of data, as follows:
 If the residual plot exhibits randomness, then a line is a good model for the data (see left)
 If the residual plot exhibits a pattern, then a line is NOT a good model for the data (right)
Outliers and Influential Points
A point that lies outside the overall pattern of the other observations is considered an outlier. If
the removal of such a point has a large effect on the correlation and/or regression, that point is
considered an influential point.
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AP Statistics Chapter 4 – Designing Studies
4.1: Surveys and Samples
Population, Census and Sample
 The population in a statistical study is the entire group of individuals we want
information about. For example, all registered voters in a given county.
 A census collects data from every individual in the population.
 A sample is a subset of individuals in the population from which we actually collect data.
Bias
The design of a statistical study shows bias if it would consistently underestimate or consistently
overestimate the value you want to know.
Convenience Sampling
A convenience sample chooses the individuals easiest to reach. This will typically result in a
biased sample of like-minded individuals.
Voluntary Response Sample
A voluntary response sample consists of people who choose themselves by responding to a
general invitation. Voluntary response samples show bias because people with strong opinions
(often in the same direction) are most likely to respond.
Simple Random Sample
A simple random sample (SRS) of size n consists of n individuals from the population chosen
in such a way that every set of n individuals has an equal chance to be the sample actually
selected.
Random Digits
A table of random digits is a long string of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 with these two
properties:
1. Each entry in the table is equally likely to be any of the 10 digits 0 through 9.
2. The entries are independent of each other. That is, knowledge of one part of the table
gives no information about any other part.
Choosing an SRS
Choose an SRS in two steps:
Step 1: Label. Assign a numerical label to every individual in the population.
Step 2: Table. Use Table B to select labels at random.
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Stratified Random Sample
To get a stratified random sample, start by classifying the population into groups of similar
individuals, called strata. Then choose a separate SRS in each stratum and combine these SRSs
to form the sample.
Cluster Sample
To get a cluster sample, start by classifying the population into groups of individuals that are
located near each other, called clusters. Then choose an SRS of the clusters. All individuals in
the chosen clusters are included in the sample.
Forms of Bias in Surveys and Samples
 Undercoverage occurs when some members of the population cannot be chosen in a
sample.
 Nonresponse occurs when an individual chosen for the sample can’t be contacted or
refuses to participate.
 A systematic pattern of incorrect responses in a sample survey leads to response bias.
 The wording of questions is the most important influence on the answers given to a
sample survey.
4.2: Experiments
Observational Study vs Experiment
 An observational study observes individuals and measures variables of interest but does
not attempt to influence the responses.
 An experiment deliberately imposes some treatment on individuals to measure their
responses.
 When our goal is to understand cause and effect, experiments are the only source of fully
convincing data. The distinction between observational study and experiment is one of
the most important in statistics.
Confounding occurs when two variables are associated in such a way that their effects on a
response variable cannot be distinguished from each other. Observational studies often fail to
provide valid causal links between variables due to confounding
The Language of Experiments
A specific condition applied to the individuals in an experiment is called a treatment. If an
experiment has several explanatory variables, a treatment is a combination of specific values of
these variables.
The experimental units are the smallest collection of individuals to which treatments are
applied. When the units are human beings, they often are called subjects.
524
Principles of Experimental Design
The basic principles for designing experiments are as follows:
1. Comparison. Use a design that compares two or more treatments.
2. Random assignment. Use chance to assign experimental units to treatments. Doing so
helps create roughly equivalent groups of experimental units by balancing the effects of
other variables among the treatment groups.
3. Control. Keep other variables that might affect the response the same for all groups.
4. Replication. Use enough experimental units in each group so that any differences in the
effects of the treatments can be distinguished from chance differences between the
groups.
Statistical Significance
 An observed effect so large that it would rarely occur by chance is called statistically
significant.
 A statistically significant association in data from a well-designed experiment does imply
causation.
Completely Randomized Design
 In a completely randomized design, the treatments are assigned to all the experimental
units completely by chance.
 Some experiments may include a control group that receives an inactive treatment or an
existing baseline treatment.
 The response to a dummy treatment is called the placebo effect.
 In a double-blind experiment, neither the subjects nor those who interact with them and
measure the response variable know which treatment a subject received.
Block Design
A block is a group of experimental units that are known before the experiment to be similar in
some way that is expected to affect the response to the treatments.
In a randomized block design, the random assignment of experimental units to treatments is
carried out separately within each block.
Matched Pairs Design
 A matched pairs design is a randomized blocked experiment in which each block
consists of a matching pair of similar experimental units.
 Chance is used to determine which unit in each pair gets each treatment.
 Sometimes, a “pair” in a matched-pairs design consists of a single unit that receives both
treatments. Since the order of the treatments can influence the response, chance is used to
determine with treatment is applied first for each unit.
525
AP Statistics Chapter 5 – Probability: What are the Chances?
5.1: Randomness, Probability and Simulation
Probability
The probability of any outcome of a chance process is a number between 0 and 1 that describes
the proportion of times the outcome would occur in a very long series of repetitions.
Simulation
The imitation of chance behavior, based on a model that accurately reflects the situation, is
called a simulation.
Performing of a Simulation – The 4-Step Process
1. State: Ask a question of interest about some chance process.
2. Plan: Describe how to use a chance device to imitate one repetition of the process. Tell
what you will record at the end of each repetition.
3. Do: Perform many repetitions of the simulation.
4. Conclude: Use the results of your simulation to answer the question of interest.
5.2: Probability Rules
Sample Space
The sample space S of a chance process is the set of all possible outcomes.
Probability Models
Descriptions of chance behavior contain two parts:
A probability model is a description of some chance process that consists of two parts:
• a sample space S and
• a probability for each outcome.
For example: When a fair 6-sided die is rolled, the Sample Space is S = {1, 2, 3, ,4,5, 6}.
The probability for a fair die would include the probabilities of these outcomes, which are all the
same.
Outcome
Probability
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
Event
An event is any collection of outcomes from some chance process. That is, an event is a subset
of the sample space. Events are usually designated by capital letters, like A, B, C, and so on.
For example: For the probability model above we might define event A = die roll is odd. The
elements of the sample space S that fits this event are {1, 3, 5}. The probability of the event A,
written as P(A) is the 3/6 or ½. So we would write P(A) = 0.5, in decimal form.
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The Basic Rules of Probability
• For any event A, 0 ≤ P(A) ≤ 1.
• If S is the sample space in a probability model, P(S) = 1.
• In the case of equally likely outcomes,
number of outcomes corresponding to event A
P( A) 
total number of outcomes in sample space
C
• Complement rule: P(A ) = 1 – P(A)
• Addition rule for mutually exclusive events: If A and B are mutually exclusive,
P(A or B) = P(A) + P(B). Also be familiar with the notation: 𝑷(𝑨 ∪ 𝑩).
Mutually Exclusive Events
Two events A and B are mutually exclusive (or disjoint) if they have no outcomes in common
and so can never occur together—that is, if P(A and B ) = 0. Alternate notation: 𝑷(𝑨 ∩ 𝑩).
For example: Using a deck of playing cards and drawing a card at random, the events A = card
is a King, and B = card is a Queen are mutually exclusive because a single card cannot be both a
King and a Queen. Thus we can calculate the probability of A or B as the sum of their individual
probabilities - P(A or B) = P(A) + P(B).
General Addition Rule
If A and B are any two events resulting from some chance process, then
P(A or B) = P(A) + P(B) – P(A and B)
Venn Diagrams and Probability
The complement Ac contains exactly the
outcomes that are not in A.
The events A and B are mutually exclusive
(disjoint) because they do not overlap. That
is, they have no outcomes in common.
The intersection of events A and B (A ∩ B)
is the set of all outcomes in both events A
and B.
The union of events A and B (A ∪ B) is the
set of all outcomes in either event A or B.
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5.3: Conditional Probability and Independence
Conditional Probability
The probability that one event happens given that another event is already known to have
happened is called a conditional probability.
Suppose we know that event A has happened. Then the probability that event B happens given
that event A has happened is denoted by P(B | A). The symbol “” is read as “given that,” so we
read P(B | A) as the probability that B occurs given that A has already occurred.
Calculating Conditional Probability
To find the conditional probability P(A | B), use the formula
P( A | B) 
P( A  B)
P( B)
The conditional probability P(B | A) is given by
P( B | A) 
P( B  A)
P( A)
The General Multiplication Rule
The probability that events A and B both occur can be found using the general multiplication rule
P(A ∩ B) = P(A) • P(B | A),
where P(B | A) is the conditional probability that event B occurs given that event A has already
occurred.
Conditional Probability and Independence
Two events A and B are independent if the occurrence of one event does not change the
probability that the other event will happen. In other words, events A and B are independent if
P(A | B) = P(A) and P(B | A) = P(B).
The Multiplication Rule for Independent Events
If A and B are independent events, then the probability that A and B both occur is
P(A ∩ B) = P(A) • P(B)
528
AP Statistics Chapter 6 – Discrete, Binomial & Geometric Random Variables
6.1: Discrete Random Variables
Random Variable
A random variable is a variable whose value is a numerical outcome of a random phenomenon.
Discrete Random Variable
A discrete random variable X has a countable number of possible values. Generally, these values
are limited to integers (whole numbers). The probability distribution of X lists the values and
their probabilities.
Value of X
x1
x2
x3
…
xk
Probability
p1
p2
p3
…
pk
The probabilities pi must satisfy two requirements:
1. Every probability pi is a number between 0 and 1.
2. p1 + p2 + … + pk = 1
Find the probability of any event by adding the probabilities pi of the particular values xi that
make up the event.
Continuous Random Variable
A continuous random variable X takes all values in an interval of numbers and is measurable.
Mean (Expected Value) of A Discrete Random Variable
Suppose that X is a discrete random variable whose distribution is
Value of X
x1
x2
x3
…
xk
Probability
p1
p2
p3
…
pk
To find the mean of X, multiply each possible value by its probability, then add all the products:
𝝁𝒙 = 𝑬(𝒙) = ∑ 𝒙𝒊 ∙ 𝒑𝒊 = 𝒙𝟏 ∙ 𝒑𝒊 + 𝒙𝟐 ∙ 𝒑𝟐 + ⋯ + 𝒙𝒌 ∙ 𝒑𝒌
529
6.3: The Binomial Distributions
A binomial probability distribution occurs when the following requirements are met.
1.
2.
3.
4.
Each observation falls into one of just two categories – call them “success” or “failure.”
The procedure has a fixed number of trials – we call this value n.
The observations must be independent – result of one does not affect another.
The probability of success – call it p - remains the same for each observation.
Notation for binomial probability distribution
n denotes the number of fixed trials
k denotes the number of successes in the n trials
p denotes the probability of success
1 – p denotes the probability of failure
Binomial Probability Formula
P( X  k ) 
n!
( p) k (1  p) n  k
k!(n  k )!
How to use the TI-83/4 to compute binomial probabilities *
There are two binomial probability functions on the TI-83/84, binompdf and binomcdf
binompdf is a probability distribution function and determines P( X  k )
binomcdf is a cumulative distribution function and determines P( X  k )
*Both functions are found in the DISTR menu (2nd-VARS)
Probability
Calculator Command
Example (assume n = 4, p = .8)
P( X  k )
binompdf(n, p, k)
P( X  3) = binompdf(4, .8, 3)
P( X  k )
binomcdf(n, p, k)
P( X  3) = binomcdf(4, .8, 3)
P( X  k )
binomcdf(n, p, k - 1)
P( X  3) = binomcdf(4, .8, 2)
P( X  k )
1 – binomcdf(n, p, k)
P( X  3) = 1 – binomcdf(4, .8, 3)
P( X  k )
1 – binomcdf(n, p, k - 1)
P( X  3) = 1 – binomcdf(4, .8, 2)
Mean (expected value) of a Binomial Random Variable
Formula:   np
Meaning: Expected number of successes in n trials (think average)
Example: Suppose you are a 80% free throw shooter. You are going to shoot 4 free throws.
For n = 4, p = .8,   (4)(.8)  3.2 , which means we expect 3.2 makes out of 4 shots, on average
530
6.3: The Geometric Distributions
A geometric probability distribution occurs when the following requirements are met.
1. Each observation falls into one of just two categories – call them “success” or “failure.”
2. The observations must be independent – result of one does not affect another.
3. The probability of success – call it p - remains the same for each observation.
4. The variable of interest is the number of trials required to obtain the first success.*
* As such, the geometric is also called a “waiting-time” distribution
Notation for geometric probability distribution
n denotes the number of trials required to obtain the first success
p denotes the probability of success
1 – p denotes the probability of failure
Geometric Probability Formula
P( X  n)  (1  p)n  1( p)
How to use the TI-83/4 to compute geometric probabilities *
There are two geometric probability functions on the TI-83/84, geometpdf and geometcdf
geometpdf is a probability distribution function and determines P( X  n)
geometcdf is a cumulative distribution function and determines P( X  n)
*Both functions are found in the DISTR menu (2nd-VARS)
Probability
Calculator Command
Example (assume p = .8, n = 3)
P( X  n)
geometpdf (p, n)
P( X  3) = geometpdf(.8, 3)
P( X  n)
geometcdf(p, n)
P( X  3) = geometcdf(.8, 3)
P( X  n)
geometcdf(p, n-1)
P( X  3) = geometcdf(.8, 2)
P( X  n)
1 – geometcdf(p, n)
P( X  3) = 1 – geometcdf(.8, 3)
P( X  n)
1 – geometcdf(p, n-1)
P( X  3) = 1 – geometcdf( .8, 2)
Mean (expected value) of a Geometric Random Variable
Formula:  
1
p
Meaning: Expected number of n trials to achieve first success (average)
Example: Suppose you are a 80% free throw shooter. You are going to shoot until you make.
For p = .8,  
1
 1.25 , which means we expect to take 1.25 shots, on average, to make first
.8
531
AP Statistics – Chapter 7 Notes: Sampling Distributions
7.1 – What is a Sampling Distribution?
Parameter – A parameter is a number that describes some characteristic of the population
Statistic – A statistic is a number that describes some characteristic of a sample
Symbols
used
Proportions
Means
Sample
Statistic
p̂
x
Population
Parameter
p

Sampling Distribution – the distribution of all values taken by a statistic in all possible samples of the same
size from the same population
A statistic is called an unbiased estimator of a parameter if the mean of its sampling distribution is equal to
the parameter being estimated
Important Concepts for unbiased estimators
 The mean of a sampling distribution will always equal the mean of the population for any sample size
 The spread of a sampling distribution is affected by the sample size, not the population size.
Specifically, larger sample sizes result in smaller spread or variability.
7.2 – Sample Proportions
7.3 – Sample Means
Choose an SRS of size n from a large population with Suppose that x is the mean of a sample from a large
population proportion p having some characteristic of population with mean  and standard deviation  .
interest.
Then the mean and standard deviation of the
sampling distribution of x are
Let 𝑝̂ be the proportion of the sample having that
𝜎
characteristic. Then the mean and standard deviation
Mean: 𝜇𝑥̅ = 𝜇
Std. Dev.: 𝜎𝑥̅ =
of the sampling distribution of 𝑝̂ are
√𝑛
Mean: 𝜇𝑝̂ = 𝑝 Std. Dev.:
With the Z-Statistic: 𝑧
=
𝜎𝑝̂ = √
𝑝(1−𝑝)
With the Z-Statistic: 𝑧
𝑛
𝑝̂−𝑝
√
𝑝(1−𝑝)
𝑛
CONDITIONS FOR NORMALITY
The 10% Condition
Use the formula for the standard deviation of p̂ only
when the size of the sample is no more than 10% of
1
the population size (𝑛 ≤ 𝑁).
10
The Large Counts Condition
We will use the normal approximation to the
sampling distribution of p̂ for values of n and p that
satisfy np  10 and n(1  p)  10 .
𝑥̅ −𝜇
=𝜎
⁄ 𝑛
√
CONDITIONS FOR NORMALITY
If an SRS is drawn from a population that has the
normal distribution with mean  and standard
deviation  , then the sample mean x will have the
normal distribution N ( ,  n ) for any sample size.
Central Limit Theorem
If an SRS is drawn from any population with mean
 and standard deviation  , when n is large
(n  30) , the sampling distribution of the sample
mean x will have the normal distribution
N ( ,  n ) .
532
AP Statistics – Chapter 8 Notes: Estimating with Confidence
8.1 – Confidence Interval Basics
Point Estimate
A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic
from a sample is called a point estimate.
The Idea of a Confidence Interval
A C% confidence interval gives an interval of plausible values for a parameter. The interval is calculated from
the data and has the form: point estimate ± margin of error
The difference between the point estimate and the true parameter value will be less than the margin of error in
C% of all samples.
The confidence level C gives the overall success rate of the method for calculating the confidence interval.
That is, in C% of all possible samples, the method would yield an interval that captures the true parameter
value.
Interpreting Confidence Intervals
To interpret a C% confidence interval for an unknown parameter, say, “We are C% confident that the interval
from _____ to _____ captures the actual value of the [population parameter in context].”
Interpreting Confidence Levels
To say that we are 95% confident is shorthand for “If we take many samples of the same size from this
population, about 95% of them will result in an interval that captures the actual parameter value.”
8.2 – Estimating a Population Proportion
Conditions for Inference about a Population Proportion
 Random Sample - The data are a random sample from the population of interest.
1
 10% Rule - The sample size is no more than 10% of the population size: 𝑛 ≤ 𝑁

10
Large Counts - Counts of successes and failures must be 10 or more: 𝑛𝑝̂ ≥ 10 and 𝑛(1 − 𝑝̂ ) ≥ 10
̂ is
Standard Error of a Sample Proportion 𝒑
√
̂(𝟏 − 𝒑
̂)
𝒑
𝒏
One-Proportion z-interval
The form of the confidence interval for a population proportion is
̂ ± 𝒛∗ √
𝒑
̂(𝟏 − 𝒑
̂)
𝒑
𝒏
Sample size for a desired margin of error
To determine the sample size (n) for a given margin of error m in a 1-proportion z interval, use formula
2
 z* 
n = p (1  p )  
m
where p* = 0.5, unless another value is given. Remember, that we will always round up to ensure a slightly
smaller margin of error than is required.
*
*
533
8.3 – Estimating a Population Mean
Conditions for Inference about a Population Mean
 Random Sample - The data are a random sample from the population of interest.
1
 10% Rule - The sample size is no more than 10% of the population size: 𝑛 ≤ 𝑁

10
Large Counts/Normality – If the sample size is large (𝑛 ≥ 30), then we can assume normality for any
shape of distribution. When sample is smaller than 30, the t procedures can be used except in the
presence of outliers or strong skewness. Construct a quick graph of the data to make an assessment.
Standard Error
When the standard deviation of a statistic is estimated from the data, the result is called the standard error of
the statistic. The standard error of the sample mean is
𝑠
√𝑛
One-Sample t-Interval for Estimating a Population Mean
The form of the confidence interval for a population mean with n  1 degrees of freedom is
𝑥̅ ± 𝑡 ∗
𝑠
√𝑛
Paired Differences t-interval
To compare the responses to the two treatments in a paired data design, apply the one-sample t procedures to
the observed differences.
For example, suppose that pre and post test scores for 10 individuals in a summer reading program are:
Subject
Pre-test
Post-test
Difference
1
25
28
3
2
31
30
-1
3
28
34
6
4
27
35
8
5
30
32
2
6
31
31
0
7
22
26
4
8
18
16
-2
9
24
28
4
We would then use the data in the “difference” row and perform one-sample t analysis on it.
534
10
30
36
6
AP Statistics – Chapter 9 Notes: Testing a Claim
9.1: Significance Test Basics
Null and Alternate Hypotheses
The statement that is being tested is called the null hypothesis (H0). The significance test is
designed to assess the strength of the evidence against the null hypothesis. Usually the null
hypothesis is a statement of "no effect," "no difference," or no change from historical values.
The claim about the population that we are trying to find evidence for is called the alternative
hypothesis (Ha). Usually the alternate hypothesis is a statement of "an effect," "a difference,"
or a change from historical values.
Test Statistics
To assess how far the estimate is from the parameter, standardize the estimate. In many
common situations, the test statistics has the form
estimate  parameter
test statistic =
standard deviation of the estimate
P-value
The p-value of a test is the probability that we would get this sample result or one more
extreme if the null hypothesis is true. The smaller the p-value is, the stronger the evidence
against the null hypothesis provided by the data.
Statistical Significance
If the P-value is as small as or smaller than alpha, we say that the data are statistically
significant at level alpha. In general, use alpha = 0.05 unless otherwise noted.
A Plan for Carrying out a Significance Test:
1.
2.
3.
4.
Hypotheses: State the null and alternate hypotheses
Conditions: Check conditions for the appropriate test
Calculations: Compute the test statistic and use it to find the p-value
Interpretation: Use the p-value to state a conclusion, in context, in a sentence or two
Type I and Type II Errors
There are two types of errors that can be made using inferential techniques. In both cases, we
get a sample that suggests we arrive at a given conclusion (either for or against H0).
Sometimes we get a bad sample that doesn’t reveal the truth.
Here are the two types of errors:
Type I – Rejecting the Ho when it is actually true (a false positive)
Type II – Accepting the Ho when it is actually false (a false negative)
Be prepared to write, in sentence form, the meaning of a Type I and Type II error in the
context of the given situation. The probability of a Type I error is the same as alpha, the
significance level. You will not be asked to find the probability of a Type II error.
535
9.2: Tests about a Population Proportion
Z-test for a Population Proportion (one-proportion z-test)
1. Hypotheses: H0: 𝑝 = 𝑝0 ; Ha: 𝑝 < 𝑝0 or 𝑝 > 𝑝0 or 𝑝 ≠ 𝑝0
2. Conditions:
o Random – does the data come from a random sample?
o Independent – is the sample size less than 10% of the population size?
o Normal – Are np0 and n(1  p0 ) both at least 10?
3. Test-Statistic: 𝑧 =
𝑝̂−𝑝0
𝑝 (1−𝑝0 )
√ 0
𝑛
where 𝑝̂ is the sample proportion
P-value: The P-value is based on a normal z-distribution. This value can be estimated
using Table A or found accurately using the 1-Prop Z-test function on your calculator
4. Conclusion: If P < , then Reject the H0, otherwise Fail to Reject H0.
9.3: Tests about a Population Mean
T-test for a Population Mean
1. Hypotheses: H0: = 0; Ha:  < 0 or  > 0 or   0
2. Conditions:
o Random – does the data come from a random sample?
o Independent – is the sample size less than 10% of the population size?
o Normal – Is it given or is there a large sample size ( n  30 )?
𝑥̅ −𝜇
3. Test-Statistic: 𝑡 = 𝑠 0where s is the sample standard deviation
⁄ 𝑛
√
P-value: The P-value is based on a t-distribution with n  1 degrees of freedom. This
value can be estimated using Table C or found accurately using the T-test function on
your calculator
4. Conclusion: If P < , then Reject the H0, otherwise Fail to Reject H0.
Paired Differences T-test
To compare the responses to the two treatments in a paired data design, apply the one-sample t
procedures to the observed differences.
For example, suppose that pre and post test scores for 10 individuals in a summer reading
program are:
Subject
Pre-test
Post-test
Difference
1
25
28
3
2
31
30
-1
3
28
34
6
4
27
35
8
5
30
32
2
6
31
31
0
7
22
26
4
8
18
16
-2
9
24
28
4
We would use the data in the differences row and perform one-sample t analysis on it.
536
10
30
36
6
AP Statistics – Chapter 10 Notes: Comparing Two Population Parameters
10.1: Comparing Two Proportions
Conditions for Comparing Two Proportions
 Random– We have two random samples, from two distinct populations
 Independence – Each sample must be selected independently of the other (no pairing or
matching) and each distinct population size must be 10 times greater than their samples.
 Normality – Counts of all “successes” and “failures” are at least 10.
Two-Proportion z Confidence Interval
To estimate the difference between two population proportions ( p1  p2 ) use the formula
𝑝̂1 (1 − 𝑝̂1 ) 𝑝̂2 (1 − 𝑝̂ 2 )
(𝑝̂1 − 𝑝̂ 2 ) ± 𝑧 ∗ √
+
𝑛1
𝑛2
Two-Proportion z-Test
To test the hypothesis H0: p1  p2 , compute the two-proportion z statistic
𝑝̂1 − 𝑝̂ 2
𝑧=
𝑝̂ (1 − 𝑝̂𝑐 ) 𝑝̂𝑐 (1 − 𝑝̂𝑐 )
√ 𝑐
+
𝑛1
𝑛2
Where 𝑝̂𝑐 =
𝑥1 +𝑥2
𝑛1 +𝑛2
given that 𝑝̂1 =
𝑥1
𝑛2
and 𝑝̂2 =
𝑥2
𝑛2
10.2: Comparing Two Means
Two-Sample Problems
 The goal of inference is to compare the responses to two treatments or to compare the
characteristics of two populations.
 We have a separate sample from each treatment or each population.
Conditions for Comparing Two Means
 Random – We have two random samples, from two distinct populations
 Independence – Each sample must be selected independently of the other (no pairing or
matching) and each distinct population size must be 10 times greater than their samples.
 Normality – Both populations are normally distributed or 𝑛1 ≥ 30 𝑎𝑛𝑑 𝑛2 ≥ 30 .
Two-Sample t Confidence Interval
To estimate the difference between two population means ( 1  2 ) use the formula
( x1  x 2 )  t
*
s12 s2 2

n1 n2
Two-Sample t-Test
To test the hypothesis H0: 1  2 , compute the two-sample t statistic
t
x1  x 2
s12 s2 2

n1 n2
537
AP Statistics Chapter 12 Inference for Distributions of Categorical Data
12.1 – Chi-Square (𝜒2) Goodness of Fit Test
Goodness of Fit
A goodness of fit test is used to help determine whether a population has a certain hypothesized
distribution, expressed as proportions of individuals in the population falling into various outcome
categories. There are two types of goodness of fit tests
1. Equal Proportions (all proportions are expected to be the same)
2. Fixed or Given Proportions (proportions are expected to follow given values)
Hypotheses for the Goodness of Fit Test
Ho: The stated distribution of the categorical variables in the population of interest is correct.
Ha: The stated distribution in the population of interest is not correct.
The Chi-Square Statistic
The formula is
(𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑐𝑜𝑢𝑛𝑡 − 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑐𝑜𝑢𝑛𝑡)2
(𝑂 − 𝐸)2
𝜒 = ∑
=∑
𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑐𝑜𝑢𝑛𝑡
𝐸
2
Expected Counts


The expected counts for the equal proportions GOF test are all the same. They are found by
dividing the total of the counts by the number of outcome categories.
The expected counts for the given proportions GOF test are NOT all the same. They are found by
multiplying the total of the counts by each given percentage.
Conditions for the Chi-Square Goodness of Fit Test



Random – The data come from a well-designed random sample or randomized experiment.
Independent – is the sample size less than 10% of the population size?
Large Counts – All expected counts are at least 5.
Degrees of Freedom for the Goodness of Fit Test
The degrees of freedom for the GOF test are n  1 where n is the number of outcome categories.
Using the Calculator for the 𝜒 2 Test on the TI-83 Calculator*




The observed counts are to be stored in L1
The expected counts are to be stored in L2
Let L3 = (L1-L2)2/L2
The 𝜒 2 is the sum of L3 (which can be found by using 1-Var Stats)
*The TI-84 has a built-in test called 𝜒 2 GOF-Test. Find it in the STAT-TEST menu.
538
12.1 – The 𝜒2 Test of Association/Independence for Two-Way Tables
Two-Way Tables
When there are two categorical variables, data can be arranged in a row and column format, called a
Two-Way Table. Here is an example:
Color Choice
Grade
1st
blue
13
green
7
red
8
yellow
2
Totals
30
2nd
11
10
6
5
33
Totals
24
17
14
7
63
Hypotheses for the Test for Association between Two Categorical Variables
Ho: There is no association between the categorical variables
Ha: There is an association
OR
Ho: The categorical variables are independent (there is no association)
Ha: The variables are NOT independent (there is an association)
Degrees of Freedom for the Chi-Square Test of Association/Independence
The degrees of freedom for the Test of Association/Independence test are (r  1) x(c  1) where r is the
number of rows and c is the number of columns in the table.
Conditions for the Chi-Square Test of Association/Independence
The conditions are the same as for the goodness of fit test
The Chi-Square Statistic
The formula is the same as in the goodness of fit test.
Expected Counts
The expected counts for this test can be found as follows:
𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑐𝑜𝑢𝑛𝑡 =
(𝑟𝑜𝑤 𝑡𝑜𝑡𝑎𝑙)(𝑐𝑜𝑙𝑢𝑚𝑛 𝑡𝑜𝑡𝑎𝑙)
𝑡𝑎𝑏𝑙𝑒 𝑡𝑜𝑡𝑎𝑙
For example, for the expected count for 2nd grade/green in the table above, we would use the
calculation
(33)(17)
= 8.9
63
539
AP Statistics Chapter 12: More about Regression
122 – Inference for Linear Regression
Sample Computer Output for a Linear Data Analysis



For the above, the linear equation is y = 7.0647 + 0.36583x
The Standard Error of the slope (SEb) = 0.01048
S = the Standard Deviation of the Residuals. Since S = 0.544, predictions of y from x
based on this regression model will be off by an average of about 0.544.
Confidence Interval for the Slope of a Regression Line
The confidence interval for β has the familiar form
statistic ± (critical value) · (standard deviation of statistic)
The t Interval for the slope 𝛽:
𝑏 ± 𝑡 ∗ 𝑆𝐸𝑏
Where b is the slope, SEb is the standard error of the slope, and t is the critical value with df = n – 2.
Performing a Significance Test for the Slope
Ho: 𝛽 = 𝛽0 (some hypothesized value – often 0)
Ha: either 𝛽 < 𝛽0 or 𝛽 > 𝛽0 or 𝛽 ≠ 𝛽0
Test Statistic: 𝑡
=
𝑏−𝛽0
𝑆𝐸𝑏
P-Value: Use the t distribution with df = n - 2
12.2 – Transformations to Achieve Linearity
Finding an Exponential Model for Data
Form: 𝒚 = 𝑨(𝑩)𝒙
Transformation: (x, log y)
Process:
1. LinReg(x, log y)
2. Resulting line is y = a + bx
3. Let A = 10a and B = 10b
Finding a Power Model for Data
Form: 𝒚 = 𝑨(𝒙)𝑩
Transformation: (log x, log y)
Process:
1. LinReg(log x, log y)
2. Resulting line is y = a + bx
3. Let A = 10a and B = b
540
Formulas for AP Statistics
I. Descriptive Statistics

x
1

 xi
n
x
i
n
x
i
x
2
n 1
y a  bx
ŷ a  bx
r
1
2
 x

i x
n 1
sx

 xi  x   yi  y 
1




n  1  sx   s y 
br
sy
sx
II. Probability and Distributions
P  A  B
 P  A  P  B   P  A  B 
Probability Distribution
P ( A | B) 
P ( A  B)
P ( B)
Mean
Standard Deviation
μ X = E ( X ) =  xi ⋅ P ( x i )
Discrete random variable, X
If X has a binomial distribution with
parameters n and p, then:
σX =
 X  np
(x

X
− μ X ) ⋅ P ( xi )
2
i
np 1  p 
n 
n x
P  X x
  x  p x 1  p 
 
where x  0, 1, 2, 3,  , n
If X has a geometric distribution with
parameter p , then:
P  X x 

where x
1  p 
x 1
X 
p
1
p
X 
1 p
p
 1, 2, 3, 
III. Sampling Distributions and Inferential Statistics
Standardized test statistic:
statistic − parameter
standard error of the statistic
Confidence interval: statistic ± ( critical value )( standard error of statistic )
2
Chi-square statistic: χ = 
( observed − expected )
2
expected
Appendix V.1 | 254
Return to Table of Contents
AP Statistics Course and Exam Description
© 2019 College Board
541
III. Sampling Distributions and Inferential Statistics (continued)
Sampling distributions for proportions:
Random Variable
For one population:
p̂
For two populations:
pˆ1 − pˆ2
Parameters of Sampling Distribution
 p̂  p
p 1  p 
 pˆ 
1
2
1
p1 1  p1 

n1
2
pˆ 1  pˆ 
s pˆ 
n
 pˆ  pˆ p1  p
 pˆ  pˆ
2
Standard Error* of
Sample Statistic
p2 1  p2 
n2
n
pˆ1 1  pˆ1 

s pˆ1  pˆ2
When
n1

pˆ 2 1  pˆ 2 
n2
p1  p2 is assumed:
1 1
s pˆ1 − pˆ2 = pˆc (1 − pˆc )  + 
 n1 n2 
X  X2
ˆc  1
where p
n1  n2
Sampling distributions for means:
Random Variable
Parameters of Sampling Distribution
For one population:
X  
X 
 X  X 1  2
 X
X
X
For two populations:
X1  X2
1
2
1
Standard Error* of
Sample Statistic

s
sX 
n
 12
n1
2

 22
s X1 
 X2
n2
n
s12 s22

n1 n2
Sampling distributions for simple linear regression:
Random Variable
Parameters of Sampling Distribution
For slope:
b  
b
b 

x n
where  x

,
Standard Error* of
Sample Statistic
sb 
x
i
x
n
s
sx n  1
2
where
and
s
sx 
,
 y
i
 2
 yi 
n2
x
i
x
2
n 1
*Standard deviation is a measurement of variability from the theoretical population. Standard error is the estimate of the standard deviation. If the standard
deviation of the statistic is assumed to be known, then the standard deviation should be used instead of the standard error.
Appendix V.1 | 255
Return to Table of Contents
AP Statistics Course and Exam Description
© 2019 College Board
542
2004 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability
Table entry for z is the
probability lying below z.
Table A
z
Standard normal probabilities
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
– 3.4
– 3.3
– 3.2
– 3.1
– 3.0
– 2.9
– 2.8
– 2.7
– 2.6
– 2.5
– 2.4
– 2.3
– 2.2
– 2.1
– 2.0
– 1.9
– 1.8
– 1.7
– 1.6
– 1.5
– 1.4
– 1.3
– 1.2
– 1.1
– 1.0
– 0.9
– 0.8
– 0.7
– 0.6
– 0.5
– 0.4
– 0.3
– 0.2
– 0.1
– 0.0
.0003
.0005
.0007
.0010
.0013
.0019
.0026
.0035
.0047
.0062
.0082
.0107
.0139
.0179
.0228
.0287
.0359
.0446
.0548
.0668
.0808
.0968
.1151
.1357
.1587
.1841
.2119
.2420
.2743
.3085
.3446
.3821
.4207
.4602
.5000
.0003
.0005
.0007
.0009
.0013
.0018
.0025
.0034
.0045
.0060
.0080
.0104
.0136
.0174
.0222
.0281
.0351
.0436
.0537
.0655
.0793
.0951
.1131
.1335
.1562
.1814
.2090
.2389
.2709
.3050
.3409
.3783
.4168
.4562
.4960
.0003
.0005
.0006
.0009
.0013
.0018
.0024
.0033
.0044
.0059
.0078
.0102
.0132
.0170
.0217
.0274
.0344
.0427
.0526
.0643
.0778
.0934
.1112
.1314
.1539
.1788
.2061
.2358
.2676
.3015
.3372
.3745
.4129
.4522
.4920
.0003
.0004
.0006
.0009
.0012
.0017
.0023
.0032
.0043
.0057
.0075
.0099
.0129
.0166
.0212
.0268
.0336
.0418
.0516
.0630
.0764
.0918
.1093
.1292
.1515
.1762
.2033
.2327
.2643
.2981
.3336
.3707
.4090
.4483
.4880
.0003
.0004
.0006
.0008
.0012
.0016
.0023
.0031
.0041
.0055
.0073
.0096
.0125
.0162
.0207
.0262
.0329
.0409
.0505
.0618
.0749
.0901
.1075
.1271
.1492
.1736
.2005
.2296
.2611
.2946
.3300
.3669
.4052
.4443
.4840
.0003
.0004
.0006
.0008
.0011
.0016
.0022
.0030
.0040
.0054
.0071
.0094
.0122
.0158
.0202
.0256
.0322
.0401
.0495
.0606
.0735
.0885
.1056
.1251
.1469
.1711
.1977
.2266
.2578
.2912
.3264
.3632
.4013
.4404
.4801
.0003
.0004
.0006
.0008
.0011
.0015
.0021
.0029
.0039
.0052
.0069
.0091
.0119
.0154
.0197
.0250
.0314
.0392
.0485
.0594
.0721
.0869
.1038
.1230
.1446
.1685
.1949
.2236
.2546
.2877
.3228
.3594
.3974
.4364
.4761
.0003
.0004
.0005
.0008
.0011
.0015
.0021
.0028
.0038
.0051
.0068
.0089
.0116
.0150
.0192
.0244
.0307
.0384
.0475
.0582
.0708
.0853
.1020
.1210
.1423
.1660
.1922
.2206
.2514
.2843
.3192
.3557
.3936
.4325
.4721
.0003
.0004
.0005
.0007
.0010
.0014
.0020
.0027
.0037
.0049
.0066
.0087
.0113
.0146
.0188
.0239
.0301
.0375
.0465
.0571
.0694
.0838
.1003
.1190
.1401
.1635
.1894
.2177
.2483
.2810
.3156
.3520
.3897
.4286
.4681
.0002
.0003
.0005
.0007
.0010
.0014
.0019
.0026
.0036
.0048
.0064
.0084
.0110
.0143
.0183
.0233
.0294
.0367
.0455
.0559
.0681
.0823
.0985
.1170
.1379
.1611
.1867
.2148
.2451
.2776
.3121
.3483
.3859
.4247
.4641
$
543
11
2004 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability
Table entry for z is the
probability lying below z.
z
Table A
(Continued)
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
.5000
.5398
.5793
.6179
.6554
.6915
.7257
.7580
.7881
.8159
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.9713
.9772
.9821
.9861
.9893
.9918
.9938
.9953
.9965
.9974
.9981
.9987
.9990
.9993
.9995
.9997
.5040
.5438
.5832
.6217
.6591
.6950
.7291
.7611
.7910
.8186
.8438
.8665
.8869
.9049
.9207
.9345
.9463
.9564
.9649
.9719
.9778
.9826
.9864
.9896
.9920
.9940
.9955
.9966
.9975
.9982
.9987
.9991
.9993
.9995
.9997
.5080
.5478
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8212
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9573
.9656
.9726
.9783
.9830
.9868
.9898
.9922
.9941
.9956
.9967
.9976
.9982
.9987
.9991
.9994
.9995
.9997
.5120
.5517
.5910
.6293
.6664
.7019
.7357
.7673
.7967
.8238
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
.9732
.9788
.9834
.9871
.9901
.9925
.9943
.9957
.9968
.9977
.9983
.9988
.9991
.9994
.9996
.9997
.5160
.5557
.5948
.6331
.6700
.7054
.7389
.7704
.7995
.8264
.8508
.8729
.8925
.9099
.9251
.9382
.9495
.9591
.9671
.9738
.9793
.9838
.9875
.9904
.9927
.9945
.9959
.9969
.9977
.9984
.9988
.9992
.9994
.9996
.9997
.5199
.5596
.5987
.6368
.6736
.7088
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
.9394
.9505
.9599
.9678
.9744
.9798
.9842
.9878
.9906
.9929
.9946
.9960
.9970
.9978
.9984
.9989
.9992
.9994
.9996
.9997
.5239
.5636
.6026
.6406
.6772
.7123
.7454
.7764
.8051
.8315
.8554
.8770
.8962
.9131
.9279
.9406
.9515
.9608
.9686
.9750
.9803
.9846
.9881
.9909
.9931
.9948
.9961
.9971
.9979
.9985
.9989
.9992
.9994
.9996
.9997
.5279
.5675
.6064
.6443
.6808
.7157
.7486
.7794
.8078
.8340
.8577
.8790
.8980
.9147
.9292
.9418
.9525
.9616
.9693
.9756
.9808
.9850
.9884
.9911
.9932
.9949
.9962
.9972
.9979
.9985
.9989
.9992
.9995
.9996
.9997
.5319
.5714
.6103
.6480
.6844
.7190
.7517
.7823
.8106
.8365
.8599
.8810
.8997
.9162
.9306
.9429
.9535
.9625
.9699
.9761
.9812
.9854
.9887
.9913
.9934
.9951
.9963
.9973
.9980
.9986
.9990
.9993
.9995
.9996
.9997
.5359
.5753
.6141
.6517
.6879
.7224
.7549
.7852
.8133
.8389
.8621
.8830
.9015
.9177
.9319
.9441
.9545
.9633
.9706
.9767
.9817
.9857
.9890
.9916
.9936
.9952
.9964
.9974
.9981
.9986
.9990
.9993
.9995
.9997
.9998
$
12
544
2004 AP® STATISTICS FREE-RESPONSE QUESTIONS
Table entry for p and
C is the point t* with
probability p lying
above it and
probability C lying
between −t * and t*.
Probability p
t*
Table B
t distribution critical values
Tail probability p
df
.25
.20
.15
.10
.05
.025
.02
.01
.005
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
1000
!
1.000
.816
.765
.741
.727
.718
.711
.706
.703
.700
.697
.695
.694
.692
.691
.690
.689
.688
.688
.687
.686
.686
.685
.685
.684
.684
.684
.683
.683
.683
.681
.679
.679
.678
.677
.675
.674
1.376
1.061
.978
.941
.920
.906
.896
.889
.883
.879
.876
.873
.870
.868
.866
.865
.863
.862
.861
.860
.859
.858
.858
.857
.856
.856
.855
.855
.854
.854
.851
.849
.848
.846
.845
.842
.841
1.963
1.386
1.250
1.190
1.156
1.134
1.119
1.108
1.100
1.093
1.088
1.083
1.079
1.076
1.074
1.071
1.069
1.067
1.066
1.064
1.063
1.061
1.060
1.059
1.058
1.058
1.057
1.056
1.055
1.055
1.050
1.047
1.045
1.043
1.042
1.037
1.036
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.303
1.299
1.296
1.292
1.290
1.282
1.282
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.684
1.676
1.671
1.664
1.660
1.646
1.645
12.71
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.021
2.009
2.000
1.990
1.984
1.962
1.960
15.89
4.849
3.482
2.999
2.757
2.612
2.517
2.449
2.398
2.359
2.328
2.303
2.282
2.264
2.249
2.235
2.224
2.214
2.205
2.197
2.189
2.183
2.177
2.172
2.167
2.162
2.158
2.154
2.150
2.147
2.123
2.109
2.099
2.088
2.081
2.056
2.054
31.82
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.423
2.403
2.390
2.374
2.364
2.330
2.326
63.66
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.704
2.678
2.660
2.639
2.626
2.581
2.576
127.3
14.09
7.453
5.598
4.773
4.317
4.029
3.833
3.690
3.581
3.497
3.428
3.372
3.326
3.286
3.252
3.222
3.197
3.174
3.153
3.135
3.119
3.104
3.091
3.078
3.067
3.057
3.047
3.038
3.030
2.971
2.937
2.915
2.887
2.871
2.813
2.807
318.3
22.33
10.21
7.173
5.893
5.208
4.785
4.501
4.297
4.144
4.025
3.930
3.852
3.787
3.733
3.686
3.646
3.611
3.579
3.552
3.527
3.505
3.485
3.467
3.450
3.435
3.421
3.408
3.396
3.385
3.307
3.261
3.232
3.195
3.174
3.098
3.091
636.6
31.60
12.92
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
4.015
3.965
3.922
3.883
3.850
3.819
3.792
3.768
3.745
3.725
3.707
3.690
3.674
3.659
3.646
3.551
3.496
3.460
3.416
3.390
3.300
3.291
50%
60%
70%
80%
90%
95%
96%
98%
99%
99.5%
99.8%
99.9%
Confidence level C
13
$
545
.0025
.001
.0005
2004 AP® STATISTICS FREE-RESPONSE QUESTIONS
Probability p
Table entry for p is the point
( χ 2 ) with probability p lying
above it.
(χ2 )
Table C
χ 2 critical values
Tail probability p
df
.25
.20
.15
.10
.05
.025
.02
.01
.005
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
80
100
1.32
2.77
4.11
5.39
6.63
7.84
9.04
10.22
11.39
12.55
13.70
14.85
15.98
17.12
18.25
19.37
20.49
21.60
22.72
23.83
24.93
26.04
27.14
28.24
29.34
30.43
31.53
32.62
33.71
34.80
45.62
56.33
66.98
88.13
109.1
1.64
3.22
4.64
5.99
7.29
8.56
9.80
11.03
12.24
13.44
14.63
15.81
16.98
18.15
19.31
20.47
21.61
22.76
23.90
25.04
26.17
27.30
28.43
29.55
30.68
31.79
32.91
34.03
35.14
36.25
47.27
58.16
68.97
90.41
111.7
2.07
3.79
5.32
6.74
8.12
9.45
10.75
12.03
13.29
14.53
15.77
16.99
18.20
19.41
20.60
21.79
22.98
24.16
25.33
26.50
27.66
28.82
29.98
31.13
32.28
33.43
34.57
35.71
36.85
37.99
49.24
60.35
71.34
93.11
114.7
2.71
4.61
6.25
7.78
9.24
10.64
12.02
13.36
14.68
15.99
17.28
18.55
19.81
21.06
22.31
23.54
24.77
25.99
27.20
28.41
29.62
30.81
32.01
33.20
34.38
35.56
36.74
37.92
39.09
40.26
51.81
63.17
74.40
96.58
118.5
3.84
5.99
7.81
9.49
11.07
12.59
14.07
15.51
16.92
18.31
19.68
21.03
22.36
23.68
25.00
26.30
27.59
28.87
30.14
31.41
32.67
33.92
35.17
36.42
37.65
38.89
40.11
41.34
42.56
43.77
55.76
67.50
79.08
101.9
124.3
5.02
7.38
9.35
11.14
12.83
14.45
16.01
17.53
19.02
20.48
21.92
23.34
24.74
26.12
27.49
28.85
30.19
31.53
32.85
34.17
35.48
36.78
38.08
39.36
40.65
41.92
43.19
44.46
45.72
46.98
59.34
71.42
83.30
106.6
129.6
5.41
7.82
9.84
11.67
13.39
15.03
16.62
18.17
19.68
21.16
22.62
24.05
25.47
26.87
28.26
29.63
31.00
32.35
33.69
35.02
36.34
37.66
38.97
40.27
41.57
42.86
44.14
45.42
46.69
47.96
60.44
72.61
84.58
108.1
131.1
6.63
9.21
11.34
13.28
15.09
16.81
18.48
20.09
21.67
23.21
24.72
26.22
27.69
29.14
30.58
32.00
33.41
34.81
36.19
37.57
38.93
40.29
41.64
42.98
44.31
45.64
46.96
48.28
49.59
50.89
63.69
76.15
88.38
112.3
135.8
7.88
10.60
12.84
14.86
16.75
18.55
20.28
21.95
23.59
25.19
26.76
28.30
29.82
31.32
32.80
34.27
35.72
37.16
38.58
40.00
41.40
42.80
44.18
45.56
46.93
48.29
49.64
50.99
52.34
53.67
66.77
79.49
91.95
116.3
140.2
14
$
546
.0025
9.14
11.98
14.32
16.42
18.39
20.25
22.04
23.77
25.46
27.11
28.73
30.32
31.88
33.43
34.95
36.46
37.95
39.42
40.88
42.34
43.78
45.20
46.62
48.03
49.44
50.83
52.22
53.59
54.97
56.33
69.70
82.66
95.34
120.1
144.3
.001
.0005
10.83
13.82
16.27
18.47
20.51
22.46
24.32
26.12
27.88
29.59
31.26
32.91
34.53
36.12
37.70
39.25
40.79
42.31
43.82
45.31
46.80
48.27
49.73
51.18
52.62
54.05
55.48
56.89
58.30
59.70
73.40
86.66
99.61
124.8
149.4
12.12
15.20
17.73
20.00
22.11
24.10
26.02
27.87
29.67
31.42
33.14
34.82
36.48
38.11
39.72
41.31
42.88
44.43
45.97
47.50
49.01
50.51
52.00
53.48
54.95
56.41
57.86
59.30
60.73
62.16
76.09
89.56
102.7
128.3
153.2
P1: OSO
FREE013-TABLE
FREE013-Moore
September 4, 2008
Random digits
LINE
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
19223
73676
45467
52711
95592
68417
82739
60940
36009
38448
81486
59636
62568
45149
61041
14459
38167
73190
95857
35476
71487
13873
54580
71035
96746
96927
43909
15689
36759
69051
05007
68732
45740
27816
66925
08421
53645
66831
55588
12975
96767
72829
88565
62964
19687
37609
54973
00694
71546
07511
95034
47150
71709
38889
94007
35013
57890
72024
19365
48789
69487
88804
70206
32992
77684
26056
98532
32533
07118
55972
09984
81598
81507
09001
12149
19931
99477
14227
58984
64817
16632
55259
41807
78416
55658
44753
66812
68908
99404
13258
35964
50232
42628
88145
12633
59057
86278
05977
05233
88915
05756
99400
77558
93074
69971
15529
20807
17868
15412
18338
60513
04634
40325
75730
94322
31424
62183
04470
87664
39421
29077
95052
27102
43367
37823
36809
25330
06565
68288
87174
81194
84292
65561
18329
39100
77377
61421
40772
70708
13048
23822
97892
17797
83083
57857
66967
88737
19664
53946
41267
28713
01927
00095
60227
91481
72765
47511
24943
39638
24697
09297
71197
03699
66280
24709
80371
70632
29669
92099
65850
14863
90908
56027
49497
71868
74192
64359
14374
22913
09517
14873
08796
33302
21337
78458
28744
47836
21558
41098
45144
96012
63408
49376
69453
95806
83401
74351
65441
68743
16853
96409
27754
32863
40011
60779
85089
81676
61790
85453
39364
00412
19352
71080
03819
73698
65103
23417
84407
58806
04266
61683
73592
55892
72719
18442
77567
40085
13352
18638
84534
04197
43165
07051
35213
11206
75592
12609
47781
43563
72321
94591
77919
61762
46109
09931
60705
47500
20903
72460
84569
547
12531
42648
29485
85848
53791
57067
55300
90656
46816
42006
71238
73089
22553
56202
14526
62253
26185
90785
66979
35435
47052
75186
33063
96758
35119
88741
16925
49367
54303
06489
85576
93739
93623
37741
19876
08563
15373
33586
56934
81940
65194
44575
16953
59505
02150
02384
84552
62371
27601
79367
42544
82425
82226
48767
17297
50211
94383
87964
83485
76688
27649
84898
11486
02938
31893
50490
41448
65956
98624
43742
62224
87136
41842
27611
62103
48409
85117
81982
00795
87201
45195
31685
18132
04312
87151
79140
98481
79177
48394
00360
50842
24870
88604
69680
43163
90597
19909
22725
45403
32337
82853
36290
90056
52573
59335
47487
14893
18883
41979
08708
39950
45785
11776
70915
32592
61181
75532
86382
84826
11937
51025
95761
81868
91596
39244
41903
36071
87209
08727
97245
96565
97150
09547
68508
31260
92454
14592
06928
51719
02428
53372
04178
12724
00900
58636
93600
67181
53340
88692
03316
Revised Pages
7:30
548
2
A z-score is the number of standard
deviations a data value is away from the
mean. The sign indicates the direction –
above or below.
Standard deviation is the average distance
between the data values and the mean.
Interpret standard 1
deviation
Interpret z-score
SOCS: Describe shape, outliers, center, and
spread (SOCS). Choose appropriate
measures of center and spread based on
shape.
What to do
Describe distribution of 1
quantitative variable
Chapter
The z-score for Bob’s test equals -1.5. Bob's score is 1.5 standard
deviations below the mean of 80.
On average, test scores are 10 points away from the mean of 80.
The distribution of test scores is unimodal and nearly symmetric.
The mean equals 80 and the standard deviation equals 10.
Example
Graph only approximates the distribution. Your graph should have axes labeled.
Scores on a stats test had a mean equal to 80 and a standard deviation equal to 10. A student named Bob scored a 65.
Scenario for examples
Quantitative Distributions
AP Statistics
Interpreting and Describing
549
Prepare side-by-side graphs. Compare
distributions, using comparison words
(higher/equal/lower).
The distribution of males scores is bimodal and skewed negative
while the distribution of female scores is unimodal and nearly
symmetric. Males had a lower mean score than females (75 vs.
84) and the standard deviation is higher for males, which
indicates their scores were less consistent.
Example
relationship
Describe 3, 12
Chapters
Describe strength, direction and form. Identify
outliers, including influential observations.
What to do
There is a strong, positive, linear association between hours
studied and test score. The point representing a student who
studied five hours and scored a 100 is an influential observation.
Example
Graph only approximates the association. Your graph should have labels.
Scores on a stats test had a mean equal to 80 and a standard deviation equal to 10. Students studied a mean of 2 hours for the test with a
standard deviation of 1. The relationship between is study time and test score is linear and the correlation equals 0.7. Bob studied 30 minutes
and scored a 65 on the test.
Scenario for examples
When both variables are quantitative
Describe relationship
1
Chapter What to do
The mean score of males on a stats test equals 75 with a standard deviation of 13. The mean score of females on the same test equals 84 and
the standard deviation equals 9.
Scenario for examples
When at least one variable is categorical
Two-variable Relationships
550
The estimated value of y when x equals 0.
Strength and direction of linear associations.
Do not use when association isn’t linear.
The percentage of <y> variance explained by
<x>
Estimated value of <y> for a given value of <x>
Difference between the true value of <y> and
the estimated value of <y>
The points in the residual plot are distributed
randomly about the x-axis. The association of
the original data is linear.
Interpret 3, 12
correlation
Interpret 3, 12
coefficient of
determination
3, 12
3, 12
Interpret y-hat
Interpret residual
Describe and 3, 12
interpret residual
plot
Rise over run: Slope is predicted amount <y>
changes by <rise amount> for per <1 unit
change> in <x>.
3, 12
Interpret y- 3. 12
intercept
Interpret slope
What to do
Chapters
Graph only approximates the association. Your graph should have
labels
The residuals are distributed randomly about the x-axis. The least
squares model is a good fit for the data.
Bob's actual score is 4.5 points lower than predicted.
The estimated grade of students who study 0.5 hours equals 69.5
49% of test score variance is explained by time spent studying
r= 0.7 means that there is a moderately strong positive
relationship between time spent studying and test score.
The estimated grade of students who don't study equals 66.
On average, test scores increase 7 points for every extra hour of
study.
Example
551
P-value
9, 10, 11,
12
Explain Type II error 12
9, 10, 11,
9, 10, 11,
Explain Type I error 12
Probability of a sample result that is at least as
extreme.
False negative
False positive
Probability that a test detects a difference
9, 10, 11,
12
Explain power
Linkage + context: Compare p-value to
significance level; reject fail/fail to reject;
answer with context
<CL%> of intervals capture the true
<parameter>
Explain confidence
8, 10, 12
level
Hypothesis test 9, 10, 11,
conclusion 12
Identify the confidence level and the
boundaries of the interval. Use context.
Confidence
8, 10, 12
interval conclusion
Chapters What to do
If male stats teachers have the same mean weight as other male
math teachers, the probability is 0.16066 that a sample of 61
male stats teachers would be 175 pounds or more.
Though male stats teachers have a higher mean weight than
other male math teachers, the test fails to detect the difference.
Though male stats teachers have the same mean weight as other
male math teachers, the test indicates male stats teachers are
heavier.
This test has a 35% probability of detecting that male stats
teachers have a mean weight that is 5 pounds heavier than the
mean weight of other math teachers. (Note: I made up the
power. It may actually be different).
Because the 𝑝 = 0.16066 is greater than 𝛼 = 0.05, do not
reject the null hypothesis. The mean weight of male stats
teachers may equal 173 pounds.
95% of intervals calculated using samples of the same size will
include the true mean weight of male stats teachers.
With 95% confidence, the mean weight of male stats teachers is
between 171 and 179 pounds.
Example
Male math teachers, other than stats teachers, have a mean weight of 173 pounds. The mean weight of a random sample of 61 male stats
teachers equals 175 pounds with a standard deviation equal to 15.6 pounds. Is this evidence that the mean weight of male stats teachers is
greater than the mean weight of other male math teachers?
Scenario for examples
Inference
552
553
554
555
556
557
558
559
560
561
2
C
I
S
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
STATS MEDIC
REVIEW
MC and FRQ
587
Unit 1 Practice Free Response Question
Below are boxplots that summarize the weights (in pounds) of large samples from two breeds of
dog: the Anatolian Shepherd and the Black Russian Terrier.
(a) Compare the distributions of weights for the two dog breeds.
(b) This sample of Black Russian Terriers does not contain any outliers. What weights would a
Black Russian Terrier have to be to be considered an outlier?
588
Unit 1 Multiple Choice
FREQUENCY
1. The graph below displays the amount of time to the nearest hour spent on homework per week
for a sample of students. Which measures of center and variability would be most appropriate to
describe the given distribution?
16
14
12
10
8
6
4
2
0
14
12
15 15
11
8
5
4 4
3
2
1 1
1 2 3 4 5 6 7 8 9 10 11 12 13
TIME (HOURS)
(A) Mean and standard deviation
(B) Mean and IQR
(C) Median and standard deviation
(D) Median and IQR
(E) Median and Range
589
2. The boxplot below summarizes the heights (in centimeters) of a sample of 200 3-year-old males in
the United States. Which of the following statements is false?
(A) 50% of the children in the sample had heights between 93 and 98 cm.
(B) 25 children had heights greater than 98 cm.
(C) The 25th percentile in this sample is a height of 93 cm.
(D) The tallest child in this sample was 111 cm tall.
(E) Every child in the sample was at least 82 cm tall.
3. A college professor calculates the standard deviation of all the grades from the midterm exams
she most recently administered. Which of the following is the best description of the standard
deviation?
(A) The difference between the highest score on the midterm and the lowest score on the midterm.
(B) The difference between the score representing the 75th percentile of all midterm exams and the
score representing the 25th percentile of all midterm exams.
(C) Approximately the mean distance between each individual grade of the midterm exams.
(D) Approximately the mean distance between the individual grades of the midterm exams and the
mean grade of all midterm exams.
(E) Approximately the median distance between the individual grades of the midterm exams and the
median grade of all midterm exams.
590
4. Consider the stemplot below, which gives the number of people attending a matinee show at the
local movie theatre for the past 26 days.
1
2
3
4
5
6
7
8
9
10
11
12
8
0
0
1
3
3
1
1
2
7
1
3
2
3
1
5
5
5 7
2 8
5 8
9
7
2
Key: 1|8 = 18 people
Which of the following statements are true?
I. The five number summary is 18, 35, 54, 87, 123
II. Half of the data values are larger than 54.
III. The shape of the distribution is skewed left.
(A) I and II
(B) II
(C) II and III
(D) III
(E) I, II, and III
5. Which of the following statements is true?
(A) Histograms have gaps between each bar.
(B) Dotplots do not provide enough information to determine if there are outliers in the data.
(C) Bar graphs can display both quantitative and categorical data.
(D) Stemplots are the best graphs for displaying data sets with two variables.
(E) Boxplots clearly show the five-number summary of a data set.
591
6. A distribution has a shape that is strongly skewed left. Which of the following statements is most
likely true about this distribution?
(A) Mean > Median.
(B) Mean = Median.
(C) Mean < Median.
(D) Only the mode can be determined.
(E) There are no outliers in the distribution.
7. Several students in an AP Statistics class administer a survey to a sample of students in their
school, asking the students how far they travel to school in miles. Which of the following statistics
would not be measured in miles?
(A) Mean
(B) Median
(C) Interquartile range
(D) Standard deviation
(E) Variance
8. The following table shows how high school students in four grade levels ate their lunch on a
particular day – purchased in the cafeteria, brought a bag lunch, or purchased lunch off campus.
What is the approximate proportion of Juniors that purchased lunch off-campus on this particular
day?
Cafeteria
Bagged Lunch
Off-Campus
Freshman
65
120
0
Sophomore
70
100
20
Junior
50
70
45
(A) 0.06
(B) 0.37
(C) 0.24
(D) 0.27
(E) 0.35
592
Senior
30
60
65
9. A club bowling team has 12 members. The first ten members bowl a game, and their average
score is 156. If you know the next two members bowl scores of 173 and 151, do you have enough
information to find the total of all 12 scores?
(A) Yes, the total is 157.
(B) Yes, the total is 1872.
(C) Yes, the total is 1884.
(D) No, we would need to know the ten individual scores.
(E) No, we would need to know the standard deviation.
10. The mosaic plot shows the distribution of favorite math class for students and teachers at East
Kentwood High School. Based on the graph, which of the following statements is true?
(A) There are more teachers than students who chose their favorite math class as AP Calc.
(B) 100% of students chose Intro Stats as their favorite math class.
(C) When combining students and teachers, AP Stats has the highest overall proportion who
chose it as their favorite math class.
(D) 80% of students chose AP Calc.
(E) There are more teachers than students at East Kentwood High School.
593
Unit 2 Practice Free Response Question
The computer output below gives results from the linear regression analysis for predicting the
pounds of fuel consumed based on the distance traveled in miles for passenger aircraft. Data used
for this analysis were obtained from ten randomly selected flights.
Predictor
Constant
Distance (miles)
Coef
-4702.64
21.282
SE Coef
1657
0.833
S = 2766.57
R-Sq = 98.8%
R-Sq(adj)=98.3%
T
-2.84
25.54
P
0.022
0.000
(a) What is the equation of the least-squares regression line that describes the relationship between
the distance traveled in miles and the pounds of fuel consumed? Define any variables used in this
equation.
(b) Below is a residual plot for the ten flights. Is it appropriate to use the linear regression equation
to make predictions? Explain.
(c) Interpret the y-intercept in the context of the problem. Is this value statistically meaningful?
594
Unit 2 Multiple Choice
1. Which of the following relationships between two variables could be described using correlation,
r?
(A) Number of books read and gender of a student.
(B) Number of football games played and the position of a football player.
(C) High temperature of the day and number of zoo visitors that day.
(D) Type of beverage ordered and time of day it was ordered.
(E) Brand of cell phone and number of cell phones sold.
2. A scatterplot shows a strong, positive, linear relationship between the number of rebounds a
basketball team averages and the number of wins that team records in a season. Which conclusion is
most appropriate?
(A) A team that increases its number of rebounds causes its chances of winning more games to
increase.
(B) If the residual plot shows no pattern, then it is safe to conclude that getting more rebounds
causes more wins, on average.
(C) If the residual plot shows no pattern, then it is safe to conclude that getting more wins causes
more rebounds, on average.
(D) If the 𝑟 ! value is close enough to 100%, then it is safe to conclude that getting more rebounds
causes more wins, on average.
(E) Rebounds and wins are positively correlated, but we cannot conclude that getting more
rebounds causes more wins, on average.
3. Two variables, x and y, have a correlation of 0.75. If x has a mean of 25 and a standard deviation
of 3, and y has a mean of 12 and a standard deviation of 6, which of the following is the leastsquares regression line for the two variables?
(A) 𝑦 = −25.5 + 1.5𝑥
(B) 𝑦 = 12 + 1.5𝑥
(C) 𝑦 = 12 + 0.75𝑥
(D) 𝑦 = 16 + 0.75𝑥
(E) Not enough information
595
4. Data are collected on the amount of fat (in grams) and calories in the french fry orders at nine fast
food restaurants. The least-squares regression line for the data is 𝑦 = 274.34 + 9.55𝑥, where 𝑦 is the
predicted number of calories and x is grams of fat. Which of the following is the correct
interpretation of the slope of the least-squares regression line?
(A) The calories increase by 9.55, on average.
(B) For every increase in fat, the calories increase as well.
(C) Every increase of 1 gram of fat causes an increase of 9.55 calories.
(D) For every increase of 1 gram of fat, the predicted calories increase by 9.55.
(E) For every increase of 1 calorie, the predicted grams of fat increase by 9.55.
5. The scatterplot below shows data for the nine french fry orders from the previous problem, with
the least-squares regression line displayed. Which of the following is the best estimate of the value
of the residual for the point indicated by the arrow?
(A) -570
(B) 570
(C) -60
(D) 60
(E) 630
596
6. The scatterplot below shows data for the nine french fry orders from the previous problem. A
tenth fast food chain has been added, as indicated by the arrow. How would this tenth data point
affect the slope and correlation in this scenario?
(A) Slope decreases, correlation increases
(B) Slope increases, correlation increases
(C) Slope increases, correlation decreases
(D) Slope decreases, correlation decreases
(E) Cannot be determined without the full set of data
7. Battery life has a strong, negative, linear relationship with temperature. If the least-squares
regression line using x = temperature explains 90% of the variation in battery life, which of the
following must be the correlation, r, between battery life and temperature?
(A) -0.90
(B) 0.90
(C) -0.95
(D) 0.95
(E) Cannot be determined without the original data.
597
8. A random sample of households is taken. For each household, the number of hours spent
watching television and the power consumption (in kWh) during a day are recorded. The table below
shows computer output from a linear regression analysis on the data.
Predictor
Constant
Hours television
Coef
19.31
0.891
SE Coef
2.8621
0.2715
S = 4.185
R-Sq = 30.0%
R-Sq(adj)=28.9%
T
6.75
3.28
P
0.000
0.002
Which of the following is the equation of the least-squares regression line?
(A) 𝑦 = 19.31 + 0.891𝑥
(B) 𝑦 = 2.8621 + 0.2715𝑥
(C) 𝑦 = 0.891 + 19.31𝑥
(D) 𝑦 = 0.2715 + 2.8621𝑥
(E) 𝑦 = 0.2715 + 0.891𝑥
9. A random sample of households is taken. For each household, the number of hours spent
watching television and the power consumption (in kWh) during a day are recorded. The table below
shows computer output from a linear regression analysis on the data.
Predictor
Constant
Hours television
Coef
19.31
0.891
SE Coef
2.8621
0.2715
S = 4.185
R-Sq = 30.0%
R-Sq(adj)=28.9%
T
6.75
3.28
P
0.000
0.002
Which of the following is a correct interpretation of r2?
(A) Number of hours of television explains 30% of the variability in power consumption.
(B) 30% of the increase in number of hours of television is explained by power consumption.
(C) 30% of the data will lie on the least-squares regression line.
(D) 30% of the residuals will be less than 4.185.
(E) All of the above are correct interpretations.
598
10. Using the least-squares regression line, ŷ = −25.5+1.5x , what is the residual for the data point at
(28,19)?
(A) -2.5
(B) 2.5
(C) 4.33
(D) 16.5
(E) 19
599
Unit 3 Practice Free Response Question
The two tires on a bicycle experience wear much in the same way car tires do; repeated contact with the road
or sidewalk will wear down the raised rubber, giving the tire less traction. A major tire manufacturer has
developed a new bicycle tire design they hope will reduce wear. After running several rounds of testing in
laboratory conditions, they want to test the tires in real-life biking conditions. They find 20 volunteers who
bike on a daily basis. The volunteers are each given a new bike of the same model for the duration of the
experiment. The volunteers are to bike as they normally would for a month, then return the bike to have the
wear of the tires measured, determined by millimeters of tire depth remaining. The manufacturer knows
these results must be compared to their standard model tire, so they will use the standard model in the
experiment as a control group.
a) The manufacturer determines a matched pair design would be most appropriate to compare the wear of
the new tire design to the standard design. Describe how to implement a matched pair design for this
experiment.
b) Describe the primary advantage of using a matched pair design in the context of this experiment.
c) At the conclusion of the experiment, the manufacturer finds that the new tire design has experienced
statistically significantly less wear than the standard tire design. Can the manufacturer infer that the new tire
will cause significantly less tire wear for all bikers? Explain.
600
Unit 3 Multiple Choice
1. An AP Statistics class starts a project to estimate the average number of hours a student at their
high school sleeps per night. Their high school has 1200 students, and they take a sample of the first
120 students that arrive at school on a particular day. They ask each of the 120 students how many
hours of sleep they got the night before and then calculate an average. Which of the following
statements is an accurate description of the elements of this survey?
(A) Population: the 120 students surveyed. Parameter of interest: the average number of hours a
student at this high school sleeps per night.
(B) Sample: the 1200 students of the school. Population: all high school students in the United
States. Statistic: the average number of hours a student at this high school sleeps per night.
(C) Sample: the 120 students surveyed. Population: the 1200 students at the high school. Parameter
of interest: the average number of hours a student at this high school sleeps per night.
(D) Sample: the 120 students surveyed. Population: the 1200 students at the high school. Statistic:
The average number of hours a student at this high school sleeps per night.
(E) This is not a survey because a convenience sample was used.
2. At a high school, the cafeteria staff provides optional comment cards to students eating in the
cafeteria. Students who wish to make a comment can obtain and fill out a comment card and return
it to a drop box in the front office. At the end of the semester, the staff analyzes the responses and
finds that 72% of the students who filled out a comment card recommend major changes to the food
selection. What conclusions can the cafeteria staff draw from these responses?
(A) Since all of the students had an opportunity to respond, this is evidence that a majority of all the
students supports major changes.
(B) The cafeteria staff should not use the results to draw any conclusions since some students may
not have filled out a comment card, which is nonresponse.
(C) If the students recorded their grade level with their response, the staff could make the original
sample a stratified sample and use the responses to learn what proportion of each grade
recommends major changes to food selection.
(D) The responses were from a voluntary response sample, so the cafeteria staff should not use the
results to draw any conclusions.
(E) The opportunity to fill out comment cards should be offered to students at all high schools in the
school district to get a larger sample size, which will make the results more reliable.
601
3. The Career Placement department at a university wants to administer a survey to determine
student awareness of the resources offered. The department expects that student awareness of the
Career Placement resources will vary significantly depending on how many years a student has
attended the university, so they plan to randomly select 30 freshmen, 30 sophomore, 30 juniors, and
30 seniors. This type of sample can be best described as a
(A) simple random sample.
(B) systematic sample.
(C) stratified sample.
(D) cluster sample.
(E) block sample.
4. Major League Baseball (MLB) has recently been evaluating the timing of various events during
games in an effort to improve the pace of a game. MLB wants to know how long a mound visit,
defined as when a coach pauses the game to visit the pitcher on the mound, takes on average. MLB
randomly selects 100 games over the course of a season, and records the length, in seconds, of
every mound visit that occurs in that game. This sample of mound visits can be best described as a
(A) simple random sample.
(B) systematic sample.
(C) stratified sample.
(D) cluster sample.
(E) block sample.
5. Twenty volunteers with high cholesterol were selected for a trial to determine whether a new diet
reduces cholesterol. The volunteers were given a low-carb, low-calorie diet. After 8 weeks of the
diet, the average cholesterol of the volunteers dropped a significant amount. This study is an
example of
(A) A controlled experiment.
(B) An uncontrolled experiment.
(C) A double blind experiment.
(D) A blocked design experiment.
(E) An observational study.
602
6. The Department of Natural Resources (DNR) is planning a study to determine whether
breakwaters help decrease erosion at a large lake. They divide the shoreline into 100-foot plots,
installing some with breakwaters and some without. However, the east and west shorelines of this
lake receive very different wave patterns due to the wind. The DNR suspects this will affect the
responses to the breakwaters. Because of this, the DNR plans to treat each shoreline as a different
group. This study is an example of
(A) An uncontrolled experiment.
(B) A blocked design experiment.
(C) A matched pairs experiment.
(D) A stratified random sample.
(E) A randomized observational study.
7. Forty adult males volunteered to participate in an experiment. Half of them are randomly assigned
to take a caffeine supplement before working out, and the other half are assigned to take a placebo
before completing the same workout. During the workout, heart rate monitors will be used to
measure each participant’s heart rate. The study found that those who took a caffeine supplement
had significantly higher average pulse rates during the workout. What conclusion can be drawn from
the study?
(A) Two experimental groups of 20 each is too small to conclude that the results are significant.
(B) Those that took the caffeine supplement had a higher average heart rate during the workout, but
it would not be correct to conclude that caffeine caused this to occur.
(C) The caffeine supplements caused the heart rates to increase due to the placebo effect.
(D) Caffeine supplements will, on average, raise the heart rates of all adult males during a workout
similar to the one performed in the experiment.
(E) Caffeine supplements will, on average, raise the heart rates of adult males similar to those in this
study during a workout similar to the one performed in the experiment.
603
8. The following 8 truck models will be used in an experiment with two treatments. The trucks are
listed according to their make and size. If the experiment uses a block design based on the size of
the trucks, which of the following is not a possible list of trucks that receive the first treatment?
Mid-size
Full-size
Chevy
Colorado
Silverado
GMC
Canyon
Sierra
Nissan
Frontier
Titan
Toyota
Tacoma
Tundra
(A) Colorado, Sierra, Frontier, Tundra
(B) Colorado, Canyon, Titan, Tacoma
(C) Colorado, Silverado, Canyon, Sierra
(D) Canyon, Frontier, Titan, Tundra
(E) Silverado, Frontier, Titan, Tacoma
9. Diverticulitis is a common intestinal ailment. To test a new medication, a hospital takes 50
volunteers who have a moderate case of the disease. The volunteers are randomly assigned to two
treatment groups, one receiving the old medication and one receiving the new medication. The pills
are distributed in unmarked pill form so that the patients and medical professionals do not know
which medication the patient is taking. Which of the following statements is/are true?
I. If the results show that the new medication is significantly more effective than the old
medication, we can conclude that the new medication will be more effective for all people
that have a moderate case of the disease.
II. The group receiving the old medication serves as a control group.
III. This is an example of a double-blind experiment.
(A) III
(B) I and II
(C) I and III
(D) II and III
(E) I, II, and III
604
10. A study is conducted to determine if the blue light from a tablet device will affect the fall asleep
time of people in various age groups differently. Volunteers for the study are grouped by age: 1830, 31-50, and 50+. Half of each group is assigned a standard tablet, the other half is assigned a
tablet with reduced blue light. People in each group are asked to use the tablet for 10 minutes
before bed and their fall asleep time is recorded. Which of the following is correct?
(A) Blocks: the three age groups. Treatments: using the standard tablet or the reduced blue light
tablet before bed. Response variable: How the eyes react to blue light.
(B) Blocks: the three age groups. Treatments: measuring how long it takes to fall asleep. Response
variable: The hours of sleep recorded.
(C) Strata: the three age groups. Treatments: using the standard tablet or the reduced blue light
tablet before bed. Response variable: How the eyes react to blue light.
(D) Blocks: the three age groups. Treatments: using the standard tablet or the reduced blue light
tablet before bed. Response variable: Time it takes to fall asleep.
(E) Blocks: the group receiving the standard tablet and the group receiving the reduced blue light
tablet. Treatments: using the standard tablet or the reduced blue light tablet before bed. Response
variable: Time it takes to fall asleep.
605
Unit 4 Practice Free Response Question
The Big River Casino is advertising a new digital lottery-style game called Instant Lotto. The player
can win the following monetary prizes with the associated probabilities:
•
5% probability to win $10
•
4% probability to win $15
•
•
3% probability to win $30
1% probability to win $50
•
0.1% probability to win the Grand Prize, $1000.
(a) Calculate the expected value of the prize for one play of Instant Lotto.
(b) As a promotion, a visitor to the casino is given 20 free plays of Instant Lotto. What is the
probability that the visitor wins some prize at least twice in the 20 free plays?
(c) The number of people who play Instant Lotto each day is approximately normally distributed with
a mean of 800 people and a standard deviation of 310 people. What is the probability that a
randomly selected day has at least 1000 people play Instant Lotto?
606
Unit 4 Multiple Choice
1. In the casino game Roulette, a bet on “red” will win if the ball lands on one of the 18 red numbers
of the 38 numbers on the wheel, with each number being equally likely. You want to run a simulation
that will estimate the probability of a player winning both bets when betting on red twice. Which of
the following would be an appropriate setup for the simulation:
I.
II.
III.
Use a table of random digits to select one number from 01 to 38 and then a second
number from 01 to 38. If the first number is between 01 and 18 and the second
number is between 01 and 18, then the player has won both rounds.
Use a table of random digits to select two numbers 01-38, without repeats. If both
numbers are between 01 and 18, then the player has won both rounds.
Use a table of random digits to select two numbers 01-19, allowing repeats. If both
numbers are between 01 and 09, then the player has won both rounds.
(A) I
(B) II
(C) III
(D) I and II
(E) I and III
2. At a summer camp, 72% of the campers participate in rope climbing and 26% participate in
canoeing. 83% of the campers participate in rope climbing, canoeing, or both. What is the
probability that a randomly selected camper participates in both rope climbing and canoeing?
(A) 0.11
(B) 0.15
(C) 0.69
(D) 0.98
(E) Cannot be determined
607
3. The table below gives information on the number of houses built in three different neighborhoods
and in three different decades. Which one of the following statements is false?
Shady Lane
Oakcrest
Pinewood Estates
1960s
40
60
0
1970s
30
15
45
1980s
10
5
15
(A) Houses in Pinewood Estates and houses built in the 1960s are mutually exclusive.
(B) The events “home is in Oakcrest” and “home was built in 1970s” are dependent.
(C) If a house is selected at random, the probability that the house was built in the 1980s is 30/220.
(D) The probability that a randomly selected home was built in the 1980s or is in the Oakcrest
neighborhood is 110/220.
(E) The probability that a randomly selected house in Shady Lane was built in the 1960s is ½.
4. According to school records, your school’s softball team wins 62% of the time when they play a
game on their home field and 26% of the time when they play at the other team’s field. This season,
they play 45% of their games at their home field. Assuming this team wins at the same pace as
previous teams, what is the probability that they win a randomly selected game this season?
(A) 0.040
(B) 0.143
(C) 0.279
(D) 0.422
(E) 0.440
5. Redundant monitoring means that a signal is sent in multiple forms or paths to reach its intended
target. For example, an alarm may be set up to send multiple signals to a security company, each
independent of each other in case of failure along one path. If an alarm sends off four signals down
different paths, and each has a 0.65 probability of reaching the security company, what is the
probability that at least one signal successfully reaches the security company?
(A) 0.015
(B) 0.111
(C) 0.178
(D) 0.821
(E) 0.985
608
6. A company advertises two car tire models. The number of thousands of miles that the standard
model tires last has a mean 𝜇( = 60 and standard deviation 𝜎( = 5. The number of miles that the
extended life tires last has a mean 𝜇. = 70 and standard deviation 𝜎. = 7. If mileages for both tires
follow a normal distribution, what is the probability that a randomly selected standard model tire will
get more mileage than a randomly selected extended life tire?
(A) 0.108
(B) 0.123
(C) 0.434
(D) 0.566
(E) 0.592
7. When playing the card game Blackjack, multiple decks are used and reshuffled often so that the
outcomes of the cards dealt are approximately independent. When a player receives two cards that
are a combination of an ace and a face card, this is called a “natural blackjack” and automatically
wins. A natural blackjack should occur in 4.5% of the rounds played. What is the probability that a
player plays 20 rounds of Blackjack and gets two or more natural blackjacks?
(A) 0.059
(B) 0.168
(C) 0.227
(D) 0.773
(E) 0.900
8. By one estimate, 3% of all Siberian Husky puppies are born with two different colored eyes (called
heterochromia iridum). For random samples of 50 Siberian Husky puppies, what are the mean and
standard deviation of the number of puppies that will have two different colored eyes?
(A) 𝜇 = 0.03, 𝜎 = 0.024
(B) 𝜇 = 0.03, 𝜎 = 0.171
(C) 𝜇 = 1.5, 𝜎 = 1.206
(D) 𝜇 = 1.5, 𝜎 = 1.455
(E) 𝜇 = 3, 𝜎 = 50
609
9. Using data collected from 1981 to 2010 for Ann Arbor, MI (GO BLUE!), the average “high”
temperature for days in July has a mean of 28.9° Celsius with a standard deviation of 3.3° Celsius.
What are the mean and standard deviation if the temperatures are converted to degrees Fahrenheit?
(A) Mean is 84.02°F, Standard deviation is 1.83°F.
(B) Mean is 52.02°F, Standard deviation is 5.94°F.
(C) Mean is 84.02°F, Standard deviation is 5.94°F.
(D) Mean is 84.02°F, Standard deviation is 35.28°F.
(E) Mean is 84.02°F, Standard deviation is 37.94°F.
10. The scores on the verbal section of the Graduate Records Examination (GRE) are approximately
normally distributed with a mean of 150 and a standard deviation of 8.5. What is the probability that
a randomly selected score on the verbal section is higher than 165?
(A) 0.018
(B) 0.039
(C) 0.961
(D) 0.982
(E) Cannot be determined
11. The finishing times of a 5K race run by competitive male runners age 15-18 are approximately
normal distribution with a mean time of 18 minutes and a standard deviation of 2 minutes. A
particularly tough 5K race tends to take these same runners 4.5 additional minutes. To earn a
completion medal in this particular 5K race, the runner must complete the race within 25 minutes.
Approximately, what percent of competitive male runners in the 15-18 age bracket typically do not
earn a completion medal at this race?
(A) Approximately 0%
(B) 10.56%
(C) 14.08%
(D) 28.93%
(E) 35.03%
610
12. A pollster asked 100 people “If money was not a factor, how many children would you like to
have?” The results and their frequencies are shown in the (estimated) probability distribution
function table below.
X
P(X)
0
?
1
2
3
4
0.26
0.29
0.15
0.05
What is the probability that the number of children a randomly selected person from this sample
would like to have is less than the mean of X?
(A) 0.25
(B) 0.49
(C) 0.51
(D) 0.80
(E) 1.49
13. Employees that work at a fish store must measure the level of nitrites in the water each day.
Nitrite levels should remain lower than 5 ppm as to not harm the fish. The nitrite level varies
according to a distribution that is approximately normal with a mean of 3 ppm. The probability that
the nitrite level is less than 2 ppm is 0.0918. Which of the following is closest to the probability that
on a randomly selected day the nitrite level will be at least 5 ppm?
(A) 0.0039
(B) 0.0266
(C) 0.0918
(D) 0.7519
(E) 0.9961
14. According to a study, 91% of all adults have a cell phone. An employee of a cell phone company
attends a community event and has a special offer to give to first time cell phone owners. If she
randomly selects adults in attendance at the event and cell phone ownership is independent from
adult to adult, what is the probability that she asks 20 adults before finding one that does not own a
cell phone?
(A) (0.91)20(0.09)
(B) (0.09)20(0.91)
(C) 20(0.91)20(0.09)
(D) 20(0.09)20(0.91)
(E) 20(0.09)(0.91)
611
15. In a large school district, it is known that 25% of all students entering kindergarten are already
reading. A simple random sample of 10 new kindergarteners is drawn. What is the probability that
fewer than three of them are able to read?
(A) 0.2503
(B) 0.2816
(C) 0.5000
(D) 0.5256
(E) 0.7759
16. A dice game at a local carnival uses a 10-sided die which has been rigged. The even numbered
outcomes (2, 4, 6, 8, and 10) have been made lighter so that they are three times more likely to
occur as the odd numbered outcomes (1, 3, 5, 7, and 9). A player that rolls an odd number wins the
game. What is the probability of winning this game?
(A) 0.05
(B) 0.15
(C) 0.20
(D) 0.25
(E) 0.50
17. Airlines must be aware of the weight of the aircraft plus everything on board. Based on
thousands of flights, the mean and standard deviation for the weights of several items are shown in
the table. The weights of the airplane, people/luggage, and fuel are assumed to be independent.
Item
Mean (in
tons)
Plane
45
People/luggage
10
Fuel
20
Standard Deviation (in
tons)
2
1
1
Let the random variable T represent the total weight of the plane, people/luggage, and fuel for a
randomly selected flight. Which of the following is closest to the standard deviation of T?
(A) 2.45 tons
(B) 4.00 tons
(C) 6.00 tons
(D) 75.0 tons
(E) 120 tons
612
18. According to the Guinness Book of World Records, a woman from Russia, Mrs. Vassilyeva, had
69 children between the years 1725 to 1765. She had 16 pairs of twins, 7 sets of triplets, and 4 sets
of quadruplets. Suppose one of the births is randomly selected. Given that Mrs. Vassilyeva gave
birth to at least 3 children (triplets), what is the probability that she gave birth to quadruplets?
(A) 4/27
(B) 7/11
(C) 3/11
(D) 3/27
(E) 4/11
19. A particular manufacturer of refrigerators has come under fire as their $1,700 LGA model has
failed for a significant portion of owners. The LGA model often failed approximately 1-2 years after
purchase, which is outside the warranty period. A class action lawsuit was filed by owners of the LGA
model claiming that the manufacturer should repair or replace the units. The lawyer representing the
class action lawsuit obtained a list of LGA owners who completed registration cards and asked a
simple random sample of 1000 LGA owners, “Is your LGA refrigerator still working?” He also
recorded the age of the refrigerator from information provided on the registration card. Here are the
data:
Working?
Yes
No
Total
Age of the LGA Refrigerator
Less than 1
1 to 2 years
More than 2
year
years
204
55
210
51
165
315
255
220
525
Total
469
531
1000
What is the probability a randomly selected LGA refrigerator is at least one year old and no longer
works?
(A) 0.051
(B) 0.135
(C) 0.480
(D) 0.531
(E) 0.904
613
20. Based on student records, 25% of the students at a large high school have a GPA of 3.5 or
better, 16% of the students are currently enrolled in at least one AP class, and 12% of the students
have a GPA of 3.5 or better and are enrolled in at least one AP class. If we select one student at
random, what is the probability that the student has a GPA lower than 3.5 and is not taking any AP
classes?
(A) 0.29
(B) 0.47
(C) 0.53
(D) 0.63
(E) 0.71
614
Unit 5 Practice Free Response Question
A regulation baseball can weigh no more than 149 grams. A factory produces baseballs with weights
that are normally distributed with a mean of 146 grams and a standard deviation of 2.3 grams.
(a) If a baseball produced by the factory is randomly selected, what is the probability that it is within
regulation weight?
(b) The baseballs are shipped in boxes of 16. What is the probability that at least 15 of the 16
baseballs in a pack are within regulation weight?
(c) The factory will not ship a box of 16 if the average weight of the baseballs in the box exceeds 147
grams. What is the probability that a pack of 16 baseballs would have an average weight of more
than 147 grams?
615
Unit 5 Multiple Choice
1. In a large city, 46% of adults support the local football team building a new stadium. If a poll is
taken from a random sample of 80 adults in the large city, which of the following properly describes
the sampling distribution of the sample proportion of adults who support the stadium?
(A) 𝜇! = 36.8, 𝜎! = 4.46, the distribution is approximately normal.
(B) 𝜇! = 36.8, 𝜎! = 4.46, shape of the distribution is unknown.
(C) 𝜇! = 0.46, 𝜎! = 0.056, the distribution is approximately normal.
(D) 𝜇! = 0.46, 𝜎! = 0.056, shape of the distribution is unknown.
(E) 𝜇! = 43.2, 𝜎! = 4.46, the distribution is binomial.
2. Which of the following statements is true?
(A) A parameter is a number that describes some characteristic of a sample.
(B) An unbiased estimator is any statistic that is taken from a sample chosen by random methods.
(C) A sampling distribution is the distribution of a statistic calculated from all possible samples of the
same size from the same population.
(D) The variability of a population distribution will decrease as the sample size increases.
(E) A normal approximation can always be used for the sampling distribution of 𝑝 as long as the
sample size is greater than 30.
3. The weight of a single bag checked by an airplane passenger follows a distribution that is right
skewed with a mean of 38 pounds and a standard deviation of 6.2 pounds. If a random sample of 96
bags is selected, what is the probability that the average weight of the bags exceeds 40 pounds?
(A) 0.0008
(B) 0.0011
(C) 0.3735
(D) 0.9992
(E) It is not appropriate to use a normal distribution to calculate probability in this situation.
616
4. Below are histograms displaying the values taken by three sample statistics in several hundred
samples from the same population. Each histogram has the same scale and the true value of the
population parameter is marked by an arrow on each histogram. Which statistic is the best estimator
of the parameter?
A
B
C
(A) Statistic A because A and B are unbiased but A has higher variability.
(B) Statistic B because A and B are unbiased but B has lower variability.
(C) Statistic C because it is always better to underestimate rather than overestimate.
(D) Statistics A or B because the population parameter is in the center of the sampling distributions
for both.
(E) Statistics A, B or C because all contain the population parameter in the sampling distributions.
5. A company intends to collect a random sample of size n from a population with population
proportion of interest p. Which of the following situations would result in the smallest standard
deviation of the sampling distribution of 𝑝?
(A) The population proportion 𝑝 = 0.5, and the sample size is 𝑛.
(B) The population proportion 𝑝 ≠ 0.5, and the sample size is 𝑛.
(C) The population proportion 𝑝 = 0.5, and the sample size is 2𝑛.
(D) The population proportion 𝑝 ≠ 0.5, and the sample size is 2𝑛.
(E) There is not enough information provided to determine the smallest standard deviation of the
sampling distribution.
617
6. Kent county and Oakland county are located on the west and east sides of Michigan, respectively.
Information about the yearly household income is given in the table below. Both distributions are
strongly skewed right. A random sample of 100 households is taken from each of the very large
populations. Which of the following describes the sampling distribution of the difference in sample
means?
Oakland County
Kent County
n
100
100
Mean (dollars)
66,400
54,340
Std. Dev. (dollars)
48,400
39,700
(A) 𝜇!! !!! = 12060, 𝜎!! !!! = 870, the distribution is approximately normal
(B) 𝜇!! !!! = 12060, 𝜎!! !!! = 6259.91, the distribution is strongly skewed right
(C) 𝜇!! !!! = 12060, 𝜎!! !!! = 6259.91, the distribution is approximately normal
(D) 𝜇!! !!! = 12060, 𝜎!!!! = 8810, the distribution is strongly skewed right
(E) 𝜇!! !!! = 12060, 𝜎!! !!! = 8810,the distribution is approximately normal
7. A recent report states that 89% of Americans consider themselves above average drivers. A local
newspaper is planning on conducting a survey to investigate whether this is true locally. If the
newspaper assumes that the 89% claim is true and plans to use the normal approximation to
calculate probabilities associated with their sample proportion, which sample size would be most
appropriate?
(A) 10
(B) 11
(C) 30
(D) 91
(E) The number depends on the size of the population.
8. Daren and Josh are pretty good free throw shooters. Daren makes 75% of the free throws he
attempts. Josh makes 80% of his free throws. Suppose we take separate random samples of 50 free
throws each from Daren and Josh, and record the proportion of free throws that are made by each.
Which of the following best describes the sampling distribution of p̂D − p̂ J ?
(A) Strong skew, with mean -0.05 and standard deviation 0.083
(B) Approximately normal, with mean -0.05 and standard deviation 0.083
(C) Shape cannot be determined, with mean of -0.05 and standard deviation 0.083
(D) Strong skew, with mean -0.05 and standard deviation 0.118
(E) Approximately normal, with mean -0.05 and standard deviation 0.118
618
9. Donner Summit, California, is a popular ski resort area. Over the past 60 years, the annual snowfall
totals of Donner Summit have followed a distribution that is strongly skewed right with a mean of
404 inches and a standard deviation of 129 inches. If many samples of size 9 were taken, which of
the following would best describe the shape of the sampling distribution of 𝑥?
(A) The shape is approximately normal since the sample size is reasonably large.
(B) The shape is skewed right since 𝑛𝑝 ≥ 10 and 𝑛(1 − 𝑝) ≥ 10 have not been met.
(C) The shape is equally as skewed right as the population distribution.
(D) The shape is skewed right but less so than the population distribution.
(E) Cannot be determined from the given information.
10. The heights of all adult males in Croatia are approximately normally distributed with a mean of
180 cm and a standard deviation of 7 cm. The heights of all adult females in Croatia are
approximately normally distributed with a mean of 158 cm and a standard deviation of 9 cm. If
independent random samples of 10 adult males and 10 adult females are taken, what is the
probability that the difference in sample means (males – females) is greater than 20 cm?
(A) 0.3463
(B) 0.6537
(C) 0.6827
(D) 0.7104
(E) 0.8687
619
Unit 6 Practice Free Response Question
How much do Americans pay for coffee on average? In a 2015 survey of 105 randomly selected U.S.
adults, the respondents were asked what price they paid for their most recent purchase of coffee.
The sample data had a mean of $3.42 and a standard deviation of $0.75.
(a) Construct and interpret a 95% confidence interval for the mean price paid for coffee in 2015 by all
U.S. adults.
(b) In 2013, tax receipts show that Americans paid an average of $2.98 per coffee. Do the 2015
survey results provide convincing evidence that the average price Americans pay for coffee has
changed? Explain your reasoning.
620
Unit 6 Multiple Choice
1. An AP Statistics class surveys 24 randomly selected female students from their high school, and
calculates a 95% confidence interval for the mean height of female students to be 63.4 ± 1.6 inches.
Which of the following is a correct interpretation of this interval?
(A) There is a 95% probability that the true mean height of female students at the school falls
between 61.8 and 65.0 inches.
(B) We can be 95% confident that the true mean height of female students at the school is 63.4
inches.
(C) 95% of the time, we can be confident that the mean sample height of female students will fall
between 61.8 and 65.0 inches.
(D) We can be 95% confident that the true mean height of female students at the school is between
61.8 and 65.0 inches.
(E) There is a 95% probability that the true mean height of female students at the school is 63.4
inches.
2. A random sample of 18 adults, chosen from the 1500 adults in the town, took a survey asking their
opinion on a recent property tax change. 25% of those who responded said they were in favor of the
change. The company running the survey wants to construct a confidence interval estimating the
proportion of all adults in the town who support the change. Which of the conditions for inference
have been satisfied?
I. Random condition
II. Normal condition
III. 10% condition
(A) I
(B) II
(C) I and III
(D) II and III
(E) I, II, and III
621
3. For which of the following samples would it be appropriate to use t-procedures for inference for
the population mean?
(A) III
(B) I and II
(C) I and III
(D) II and III
(E) I, II, and III
4. The students of a Statistics class want to estimate how many years it takes for a university
professor to earn a PhD. They survey a random sample of 40 professors with PhDs, which results in a
sample mean of 5.4 years and a standard deviation of 1.6 years. Which of the following represents
the 95% confidence interval for the true mean of the number of years it takes a professor to earn a
PhD?
(A) 5.4 ± 1.96(1.6)
(B) 5.4 ± 1.96
(C) 5.4 ± 1.96
(D) 5.4 ± 2.09
(E) 5.4 ± 2.09
+.,
-.
+.,
√-.
+.,
-.
+.,
√-.
5. A study intends to estimate a population mean with an unknown population standard deviation
and a sample size of 15. Which of the following is closest to the appropriate critical value to create a
98% confidence interval?
(A) 2.055
(B) 2.249
(C) 2.264
(D) 2.602
(E) 2.624
622
6. A newspaper plans to conduct a survey for the upcoming presidential election in order to estimate
the proportion of the population, p, who supports a certain candidate. What is the smallest sample
size needed to obtain an estimate that is within 4% of the true proportion p at the 96% confidence
level?
(A) 26
(B) 376
(C) 601
(D) 660
(E) Cannot be determined from information given.
7. A random sample of size n is collected from a considerably larger population of size N. That
sample is used to create a 95% confidence interval to estimate a population proportion. Using the
same sample proportion, the confidence interval would be narrower if:
(A) A smaller sample size was used.
(B) A t-procedure was used instead of a z-procedure.
(C) A higher confidence level was used.
(D) The population size N was larger.
(E) A lower confidence level was used.
8. Electric vehicles make up a very small proportion of the overall car market, but by how much has
that proportion increased? Independent surveys of randomly selected car dealerships were
completed, with 0.1% of the 10,000 cars sold in the 2012 sample being electric vehicles, and 0.5%
percent of the 20,000 cars in the 2016 sample being electric vehicles. Which of the following
represents a 95% confidence interval for the change in the proportion of electric cars sold from 2012
to 2016?
(A) (0.005 − 0.001) ± 1.964
(B) (0.005 − 0.001) ± 1.654
....5(+6....5)
....5(+6....5)
(C) (0.005 − 0.001) ± 1.964
(D) (0.004) ± 1.964
7.,...
7.,...
+
....+(+6....+)
+
....+(+6....+)
....:;(+6....:;)
7.,...
+.,...
+.,...
+
....:;(+6....:;)
+.,...
....-(+6....-)
:.,...
....:;(+6....:;)
(E) (0.0037) ± 1.964
:.,...
623
9. A recent study examining the effects of sugar consumption on a middle school student’s ability to
focus on a reading assignment used 18 volunteer subjects and divided them into 9 pairs based on
their reading speeds. One randomly assigned member of each pair was given a beverage containing
a substantial amount of sugar, and the other drank a sugar-free version of the beverage. Each
subject was given a passage to read and the time (in seconds) it took to read was recorded. The
difference for each pair is calculated (sugar – sugar-free). A 90% confidence interval for the mean
difference in reading times is (-5.8, 0.15).
(A) Because the center of the interval is -2.825, we have convincing evidence that sugar causes faster
reading times, on average.
(B) Because the confidence interval includes 0, we don’t have convincing evidence that sugar causes
faster reading times, on average.
(C) Because the confidence interval includes 0, we have convincing evidence that sugar causes faster
reading times, on average.
(D) Because the confidence interval includes more negative than positive values, we have convincing
evidence that sugar causes faster reading times, on average.
(E) Causation should not be inferred because the subjects were volunteers.
10. A social media developer wants to determine if the proportion of teenagers who use Facebook is
the same as the proportion of teenagers who use Snapchat. She takes a random sample of 100
teenagers and finds that 75 of the 100 students use Facebook and 89 of the 100 students use
Snapchat. Would it be reasonable for the social media developer to construct a 95% confidence
interval for the true difference in proportion of teenagers that use Facebook and Snapchat?
(A) No, the random condition is not satisfied.
(B) No, the normal condition is not satisfied.
(C) No, the two samples are not independent.
(D) Yes, all conditions have been met.
(E) Cannot be determined from the given information.
624
11. A political polling organization conducted a survey by selecting 100 random samples, each
consisting of 500 registered voters. The registered voters in each sample were asked whether they
planned to vote for the Republican or Democratic candidate in the next presidential election. For
each of the 100 samples, the polling organization created a 99 percent confidence interval for the
proportion of all registered voters who planned to vote for the Republican candidate. Which of the
following statements is the best interpretation of the 99 percent confidence level?
(A) We would expect about 99 of the 100 confidence intervals to contain the sample proportion of
the registered voters who plan to vote for the Republican candidate.
(B) We would expect about 99 of the 100 confidence intervals to contain the proportion of all
registered voters who plan to vote for the Republican candidate.
(C) We would expect the margin of error to be 0.01 because the polling organization constructed 99
percent confidence intervals.
(D) We would expect the margin of error to be less than 0.01 because the polling organization
constructed 100 different 99 percent confidence intervals.
(E) We would expect only 1 of the 100 confidence intervals to reveal that the Democratic candidate
is favored.
12. A pharmaceutical company claims that side effects will be experienced by fewer than 20% of the
patients who use medication X. A clinical trial with a random sample of 400 patients was conducted
in which half of the patients were randomly assigned to take medication X and the other half
received a placebo. Of those that received medication X they find 68 who experienced side effects.
Assuming all conditions for inference were met, which of the following is a 90 percent confidence
interval for the proportion of patients that will experience side effects while taking medication X?
(A) (0.1391, 0.2009)
(B) (0.2744, 0.4057)
(C) (0.2849, 0.3951)
(D) (0.6257, 0.7343)
(E) (0.6416, 0.7184)
625
13. Household income in the United States is strongly skewed to the right. The current presidential
administration claims that the mean household income is greater than it has ever been in the past.
An independent contractor will obtain a random sample of 100 households in the United States and
will calculate the mean household income. Which of the following statements is true?
(A) The sampling distribution of the sample mean household income is approximately normal
because the sample size of 100 is greater than 30.
(B) The distribution of household income for the sample is approximately normal because the
sample size of 100 is greater than 30.
(C) The sampling distribution of the sample mean household income is strongly skewed to the right
because the population standard deviation is unknown.
(D) The distribution of household income is strongly skewed to the left because the population
sample size of 100 is greater than 30.
(E) The sampling distribution of the sample mean household income is strongly skewed to the right
because the population distribution is strongly skewed to the right.
14. In a month-long randomized comparative experiment, participants were assigned to one of two
treatments. 35 of the 50 participants who received the first treatment, a fitness watch and training on
how to meet their daily goals as measured by the watch, lost weight. However, only 16 of the 40
participants who received the second treatment, instruction to recite positive affirmations each
morning, lost weight. Which of the following is an appropriate margin of error for a 99 percent
confidence interval to estimate the difference in the population proportion of all users of each
treatment who would lose weight?
(..;)(..:)
(A) 2.5764
(B) 1.9604
5.
(..;)(..:)
5.
+
(..-)(..,)
+
(..-)(..,)
-.
-.
(..5+)(..-=)
(C) 2.5764
+..
(..5+)(..-=)
(D) 1.9604
+..
(..;)(..:)
(E) 2.576 >
5.?
+
(..-)(..,)
-.?
@
626
15. As the temperature rises in Chicago, does the crime rate also rise? Using data available from the
Chicago Police Department, an interested citizen recorded the high temperature and number of
crimes reported for 8 randomly selected days.
Temperature ℉
17
35
46
55
64
78
84
89
Number of Crimes
56
60
66
70
71
78
74
76
The citizen wants to find a confidence interval that can be used to estimate the number of additional
crimes that can be expected to be reported for each degree that the daily high temperature
increases with 95% confidence. Which of the following is the most appropriate procedure for such an
investigation?
(A) A chi-square test of association
(B) A linear regression t-interval for slope
(C) A one-sample t-interval for a mean
(D) A two-sample t-interval for a difference of means
(E) A one-sample z-interval for a proportion
16. A large-sample 95 percent confidence interval for the proportion of credit card customers that
have reported fraudulent charges on their account is (0.028, 0.086). What is the point estimate for
the proportion of all credit card customers that have reported fraudulent charges on their account?
(A) 0.057
(B) 0.058
(C) 0.029
(D) 0.114
(E) 0.196
627
17. A writer for a career magazine is working on an article about the projected career earnings for
college graduates in various fields. He selects a random sample of 25 educators and surveys them
about their current salary, the number of years they have worked in the field, and their pay raise
structure. Based upon this information he computes the average projected career earnings for
graduates with degrees in education to be $2.5 million dollars with a standard deviation of 0.4
million dollars. Assuming all conditions for inference are met, which of the following is a 90 percent
confidence interval for the mean projected career earnings for graduates with degrees in education?
(A) 2.5 ± 1.711
..-
√75
(B) 2.5 ± 1.645(0.04)
(C) 2.5 ± 1.960
(D) 2.5 ± 1.645
..-
√75
..√75
(E) 2.5 ± 1.960(0.04)
18. A recent article claimed that women are waiting longer to have their first child. The article
estimates that the average age of first-time mothers is 26 years old, which is up from 21 years old in
1970. The margin of error for the estimate was 1.5 years. Based on the estimate and the margin of
error, which of the following is an appropriate conclusion?
(A) 95% of the women in the study were 26 years old when they had their first child.
(B) The age of every first-time mother in the sample must have been between 24.5 and 27.5 years
old.
(C) The age of every first-time mother in the sample must have been between 23 and 29 years old.
(D) This study proves that women are waiting longer to have their first child.
(E) It is plausible that the average age of first-time mothers is 27 years old.
628
19. James has a desk job and would like to become more fit, so he purchases a tread walker and a
standing desk which will allow him to walk at a slow pace as he works. However, he is concerned that
standing and walking while working may cause his productivity to decline. After working this way for
6 months he takes a simple random sample of 15 days. He records how long he walked that day (in
hours) as recorded by his fitness watch as well as his billable hours for that day as recorded by a work
app on his computer.
Regression Analysis: Billable hours versus Walk time
Predictor
Constant
Walk time
Coef
7.785
–0.245
S = 5.051
SE Coef
0.542
0.205
R-Sq = 84.3%
T
14.363
–1.195
P
0.000
0.127
R-Sq(adj) = 82.4%
Assuming that all conditions for inference are met, which of the following is a 95 percent confidence
interval for the average change in the number of billable hours for each increase of 1 hour spent
walking?
(A) –0.245 ± 1.960(0.205)
(B) –0.245 ± 2.131(0.205)
(C) –0.245 ± 2.160(0.205)
(D) 7.785 ± 1.960(0.542)
(E) 7.785 ± 2.160(0.542)
20. A curious student wanted to determine if there was a difference in the average price of a quarter
pound hamburger in the United States and Japan. The student randomly selected 15 McDonald’s
restaurants in the United States and 10 McDonald’s restaurants in Japan and recorded the prices of
their quarter pound hamburgers. Prices of the quarter pound hamburgers in Japan were converted
to U.S. dollars. The data are summarized in the table below.
Sample mean
Sample standard deviation
Sample size
U.S.
4.53
0.24
15
Japan
4.01
0.38
10
Calculate a 99 percent confidence interval for the difference in mean price for a quarter pound
hamburger in the United States and Japan.
(A) (4.53 − 4.01) ± 3.0124
..7- ?
(B) (4.53 − 4.01) ± 3.0124
..7- ?
+5
+5
+
−
..:B?
+.
..:B?
+.
(..7-)(..;,)
(C) (0.24 − 0.38) ± 3.012 >
+5?
(D) (0.24 − 0.38) ± 3.0124
(..7-)(..;,)
(E) (4.53 − 4.01) ± 3.0124
(..7-)(..;,)
+5
+5
+
(..:B)(..,7)
+
(..:B)(..,7)
+
(..:B)(..,7)
+.?
@
+.
+.
629
Unit 7 Practice Free Response Question
An online streaming service providing television programs claims that a 30-minute program will
stream with advertisements that average 45 seconds. A consumer advocacy group is investigating to
see if this claim is true. They recorded the times of 21 randomly selected advertisements. The times
are listed below:
Time
(seconds) 45 45 43 50 50 45 43 50 45 49 46 48 42 46 44 52 48 45 46 50 48
The mean and standard deviation for these times are 46.67 seconds and 2.78 seconds respectively.
Do these data provide convincing evidence that the true mean advertisement length is longer than
45 seconds?
630
Unit 7 Multiple Choice
1. A Statistics class from a high school with 4,000 students took a survey of the first 35 students who
walked through the front door of the school, and asked how far they traveled to school that day. The
class plans to run a one-sample t-test to determine if the average travel distance has increased since
last year. The students notice that the sample data are right skewed. Which conditions have been
satisfied for the t-test?
I.
II.
III.
The sample is from a random sample or randomized experiment.
The sampling distribution of sample means is approximately normal.
The sample size is small relative to the population.
(A) III
(B) I and II
(C) I and III
(D) II and III
(E) I, II, and III
2. An advertiser wants to find convincing evidence that television viewers remember more than 4
commercials, on average, after watching a 30 minute TV program. They take a random sample of
100 television viewers and ask them how many commercials they could remember after watching a
30 minute TV program. The appropriate t-test was conducted, which resulted in a P-value of 0.15.
Assuming all conditions were met, which of the following is an appropriate conclusion?
(A) Because the P-value is less than 0.05, at the 5% significance level, there is not convincing
evidence that television viewers remember more than 4 commercials, on average, after watching a
30 minute TV program.
(B) Because the P-value is greater than 0.05, at the 5% significance level, there is convincing
evidence that television viewers remember fewer than 4 commercials, on average, after watching a
30 minute TV program.
(C) Because the P-value is greater than 0.01, at the 1% significance level, there is not convincing
evidence that television viewers remember more than 4 commercials, on average, after watching a
30 minute TV program.
(D) Because the P-value is greater than 0.01, at the 1% significance level, there is convincing
evidence that television viewers remember exactly 4 commercials, on average, after watching a 30
minute TV program.
(E) Because the P-value is less than 0.01, at the 1% significance level, there is convincing evidence
that television viewers remember fewer than 4 commercials, on average, after watching a 30 minute
TV program.
631
3. After completing a statistical analysis of a survey of 40 students, the principal of North High
School made the following conclusion: reject the null hypothesis; there is convincing evidence that
more than 50% of students support a schedule change to have lunch occur earlier in the day. Which
error could have been committed?
(A) Type I error: Conclude that more than 50% of students want earlier lunch, when 50% or less want
earlier lunch.
(B) Type I error: Conclude that more than 50% of students want earlier lunch, when more than 50%
want earlier lunch.
(C) Type II error: Fail to reject that 50% of students want earlier lunch, when more than 50% want
earlier lunch.
(D) Type II error: Fail to reject that 50% of students want earlier lunch, when 50% or less want earlier
lunch.
(E) Type II error: Fail to reject that more than 50% of students want earlier lunch, when 50% or less
want earlier lunch.
4. A significance test was conducted using the hypotheses !" : $% − $' = 0, !+ : $% − $' < 0 where
$% is the true mean number of fouls called during games played at neutral sites and $' is the true
mean number of fouls called during games played at the home team’s stadium with a resulting Pvalue of 0.24. Which of the following is an accurate interpretation of this P-value?
(A) There is a 24% probability that the null hypothesis is true.
(B) If the test were repeated many times, we would correctly reject the null hypothesis 24% of the
time.
(C) If the test were repeated many times, we would incorrectly fail to reject the null hypothesis 24%
of the time.
(D) If the null hypothesis is true, there is a 24% probability of getting a sample difference in means as
far or farther below 0 as the difference found in the samples.
(E) If the null hypothesis is false, there is a 24% probability of getting a sample difference in means
as far or farther below 0 as the difference found in the samples.
632
5. Is the proportion of adults who watch the nightly news dropping? In a survey taken in 2013, 24 out
of 40 adults surveyed responded that they had watched the local TV news at least once in the last
month. In a similar survey in 2010, 40 out of 50 adults said they had watched the local TV news at
least once in the last month. Is this convincing evidence that the proportion of adults watching the
local TV news dropped between 2010 and 2013? For a significance test with the hypotheses !" : -. −
-/ = 0, !+ : -. − -/ < 0, where -. is the proportion of all adults who watched the local news at least
once a month in 2013, and -/ is the proportion for 2010, which of the following is closest to the
standard error for the test statistic?
(A)1
".3..(.5".3..)
7"
(B) 1
".:(.5".:)
(C) 1
(D)1
7"
+
+
9"
".:(.5".:)
(E) 1
+
<"
7"
7"
9"
".;(.5".;)
".3..(.5".3..)
".:(.5".:)
".3..(.5".3..)
+1
.
".3..(.5".3..)
<"
".;(.5".;)
9"
9"
.
.
".;(.5".;)
−1
.
.
6. A random sample of 95 vehicles is taken from a large parking lot at an office park. Below is the
type of each vehicle, and whether it is owned or leased by the driver. A chi-square test will be
conducted to determine if there is an association between type of vehicle and method of obtaining
the vehicle. Which of the following expressions represents the expected count of leased SUVs?
Owned
Leased
Car
29
21
SUV
20
10
Truck
11
4
(A) 10
(B)
(C)
(D)
(E)
."
<9
(=9)(=")
<9
(."5....)>
....
(....5.")>
."
633
7. A random sample of 300 balls hit into play in college baseball games is taken. For each ball hit
into play, it is recorded whether the ball was hit to the left, center, or right of the field, and whether
the ball resulted in a hit or an out. Which of the following is the appropriate null hypothesis to test if
there is a relationship between the direction of the ball and if it resulted in a hit or an out?
(A) !" : The proportion of balls hit to the left, right, and center is the same.
(B) !" : The proportion of balls that result in hits and outs is the same.
(C) !" : The distribution of the direction of the ball is the same for the population of hits and the
population of outs.
(D) !" : The direction of the ball and whether it results in a hit or an out are independent.
(E) !" : There is an association between the direction of the ball and whether it results in a hit or an
out.
8. Do stain-polyurethane mixes protect wood as well as stain and polyurethane applied in separate
coats? Five types of wood will be used, with two boards of each type of wood. One board of each
type of wood (randomly selected) will have the stain-polyurethane mix applied to it, and the other
will have stain and polyurethane applied in separate coats. Each board will have water poured on it,
and the amount of water retained will be measured. Which significance test is most appropriate?
(A) A chi-square test for goodness of fit
(B) A one-sample z-test for a proportion
(C) A matched-pair t-test for a mean difference
(D) A two-sample t-test for a difference between two means
(E) A two-sample z-test for a difference between two proportions
9. An analysis of 8 used trucks listed for sale in the 48076 zip code finds that the power model
?@(-ABCD
E ) = 3.748 − 0.1395?@(MN?DO), for price (in thousands of dollars) and miles driven (in
thousands), is an appropriate model of the relationship. If a used truck has been driven for 47,000
miles, which of the following is closest to the predicted price for the truck?
(A) $9.46
(B) $24.80
(C) $3,210.00
(D) $9,460.00
(E) $24,800.00
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10. Below is computer output from the least squares regression analysis on the body mass index,
BMI, and percent body fat for 10 randomly selected adult males. Which of the following represents
the 95% confidence interval for the slope of the regression line relating BMI and percent body fat for
the population of adult males?
Predictor
Constant
BMI
Coef
-20.096
1.695
SE Coef
2.786
0.2280
S = 3.195
R-Sq = 87.3%
R-Sq(adj)=86.8%
T
-7.213
7.432
P
0.000
0.000
(A) −20.096 ± 2.21(2.786)
(B) 1.695 ± 7.432(0.2280)
(C) 1.695 ± 2.306(0.2280)
(D) 1.695 ± 2.262(0.2280)
(E) 1.695 ± 2.228(0.2280)
11. A 20-ounce soda bottle is supposed to contain 20 ounces of soda. The distribution of the actual
amount of soda in a 20-ounce soda bottle is approximately normal. A curious student randomly
selects fifteen 20-ounce soda bottles from various retailers and carefully measures their content to
see if the soda company is cheating the customers. The mean and standard deviation of the fifteen
bottles is 19.4 ounces and 0.25 ounces, respectively. Which of the following is the test statistic for
the appropriate test to determine if the average amount of soda that is contained in their 20-ounce
soda bottles is significantly less than 20-ounces?
(A) S =
(B) X =
(C) S =
(D) X =
(E) X =
.<.75/"
T.>U
√WU
.<.75/"
1(T.>U)(WYT.>U)
WU
/"5.<.7
Z("./9)(.5"./9)
/"5.<.7
T.>U
√WU
.<.75/"
W W
]
WU >T
1("./9)(".39)[ \
635
12. A study conducted by a professor at Waynesburg University investigated whether time
perception is impaired when completing the last example problem just prior to the start of fall break.
The professor timed how long it took to complete the last example problem (in seconds), then asked
a random sample of 10 students to estimate how long it took to complete the problem (in seconds).
Let $ represent the average difference in time (actual time – estimated time) it took to complete the
problem for all students. She would like to investigate whether there will be convincing statistical
evidence that the students overestimate the time it took to complete the problem, on average.
Which of the following is the correct hypotheses for this test?
(A) H0: $
(B) H0: $
(C) H0: $
(D) H0: $
(E) H0: $
= 0 and Ha: $
= 0 and Ha: $
= 0 and Ha: $
> 0 and Ha: $
< 0 and Ha: $
≠0
>0
<0
<0
=0
13. A 21-year old college student submits a photo to the website “Guess My Age”. At a later date
he checks back and thousands of users have made guesses about his age. Of the following, which is
the best procedure to investigate whether there is convincing statistical evidence that, on average,
he is perceived to be less than 21 years old?
(A) One sample t-test for a mean
(B) One sample z-test for a proportion
(C) Matched-pairs t-test for a mean difference
(D) Two-sample t-test for the difference between two means
(E) A chi-square test of association
14. Students in a statistics class would like to investigate if more than two-thirds of the Earth is water.
To answer this question, students tossed an inflatable globe back and forth. After each catch, the
student recorded whether the tip of their pointer finger of their right hand was on water or land. In
50 tosses, their finger was on water 38 times. Assuming all conditions for inference are met, do the
data provide convincing statistical evidence at the significance level of ^ = 0.05 that more than twothirds of the Earth is water?
(A) Yes, because the p-value of 0.76 is greater than the significance level of 0.05.
(B) Yes, because the p-value of 0.08 is greater than the significance level of 0.05.
(C) Yes, because the p-value of 0.05 is less than the significance level of 0.08.
(D) No, because the p-value of 0.08 is greater than the significance level of 0.05.
(E) No, because the p-value of 0.03 is less than the significance level of 0.05.
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15. According to the National Association of Colleges and Employers, the mean salary for a new
college graduate is $45,327. A small college wants to know if the mean salary of their most recent
graduates is greater than $45,327. A random sample of 10 recent graduates from the small college
was selected and the mean and standard deviation of the salary for those graduates was found. With
all conditions for inference met, a significance test was conducted and a p-value of 0.045 was
obtained. Which of the following statements is the most appropriate conclusion using a significance
level of α = 0.05?
(A) There is convincing statistical evidence that the mean salary for all recent college graduates from
the small college is greater than $45,327.
(B) There is convincing statistical evidence that the mean salary for the sample of 10 college
graduates from the small college is greater than $45,327.
(C) There is not convincing statistical evidence that the mean salary for all recent college graduates
from the small college is greater than $45,327.
(D) There is not convincing statistical evidence that the mean salary for the sample of 10 college
graduates from the small college is greater than $45,327.
(E) There is not convincing statistical evidence that the graduates from this college will always earn
more than $45,327.
16. A financial analyst wants to investigate the relationship between the annual starting salary of new
employees at a large firm versus years of education the employee has (beyond high school). The
analyst selected a random sample of 18 new employees and recorded their starting salary and how
many years of education they have beyond high school. The computer output of an analysis of salary
versus years of education is shown in the table.
Regression Analysis: Salary (in $1,000) versus Education
Predictor
Constant
Education
Coef
25.840
13.4862
SE Coef
3.803
0.5310
T
***
***
P
***
***
S = 5.051
R-Sq
84.3%
= 82.4%is the appropriate test
Assuming that all conditions
for inference
are=met,
which R-Sq(adj)
of the following
statistic for testing the null hypothesis that the slope of the population regression line equals 0?
(A) 0
(B) 0.163
(C) 1
(D) 1.54
(E) 25.40
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17. At the county fair there are many game booths. One booth has a prize
wheel with 8 equal sectors: 2 of which are red, 2 of which are white, 2 of
which are yellow, and 2 of which are green. It costs $1 to spin the wheel.
There are 4 possible “prizes”. If the player lands on one of the green
sectors, they win $5. If they land on a yellow sector, they win a small emoji
stuffed toy. If they land on a white sector, they get a ticket to ride the Ferris
wheel. If they land on a red sector, they win nothing. Because you suspect
that the wheel is not fair, you watch a random sample of 100 people play
the game. Here are the findings:
Color
Red
White
Green
Yellow
Observed
48
22
5
25
Expected
25
25
25
25
A chi-square goodness of fit test was conducted to determine whether the data provide convincing
evidence that the wheel is not fair. The test statistic was 37.52. Which statement is true?
(A) At the significance level ^ = 0.05, the data do not provide convincing evidence that the wheel is
not fair.
(B) At the significance level ^ = 0.05, the data provide convincing evidence that the wheel is not fair.
(C) No valid conclusion can be made because the observed frequency for one cell is 5.
(D) The chi-square statistic has 100 – 1 = 99 degrees of freedom.
(E) The wheel may not be fair, but the game is fair because all of the expected values are equal.
18. A credit card company claims that the mean time the customers spend on hold is 3.5 minutes.
An employee of this company believes that customers spend more than 3.5 minutes on hold. A
random sample of n = 36 calls is selected and the mean time the customers in this sample spent on
hold was 5.15 minutes. All conditions for inference were met and the p-value for the appropriate
hypothesis test was 0.031. Which of the following statements is the best interpretation of the pvalue?
(A) There is a 0.031 probability that the alternative hypothesis is true.
(B) There is a 0.031 probability that the null hypothesis is true.
(C) If the null hypothesis is true, there is a 0.031 probability of finding convincing evidence that 5.15
minutes is greater than 3.5 minutes.
(D) If the null hypothesis is true, the probability of finding convincing evidence that the null
hypothesis is true is 0.031.
(E) If the null hypothesis is true, the probability of observing a sample mean of at least 5.15 minutes
is 0.031.
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19. A machine that makes dimes is adjusted to make them at diameter 1.9 cm. Production records
show that when the machine is properly adjusted, it will make dimes with a mean diameter of 1.9 cm
and with a standard deviation of 0.1 cm. During production, an inspector checks the diameters of
dimes to see if the machine has slipped out of adjustment. A random sample of 64 dimes is
selected. The inspector would like to detect if the true mean diameter happens to reach 1.95 cm at a
significance level of a = 0.01. He determines the power of this test to be 0.9228. What is the
probability that the inspector will make a Type II Error?
(A) 0.01
(B) 0.0385
(C) 0.05
(D) 0.0772
(E) 0.9228
20. An orange juice manufacturer advertises a new orange juice product that contains 50% less
sugar. This new product is expected to increase profits substantially because they create the 50%
less sugar product by replacing 50% of the juice with water while selling it for the same price. An
inspector wants to investigate whether the actual percentage of water in the product is within 5% of
the intended 50%. He selects a random sample of 500 containers of orange juice and determined
the percentage of each that is water. He plans to conduct a significance test at the ^ = 0.01 level to
see if there is convincing evidence that the proportion of water added is different from 0.50. What is
the probability that the inspector makes a Type I error?
(A) 0.01
(B) 0.05
(C) 0.10
(D) 0.45
(E) 0.50
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