Name ________________________________
AP Statistics
Class Notes &
Reading Guide
TPOSe5
Chapter P: Preliminary
Key Vocabulary:
statistics
probability
available data
census
observational study
sample
variable
quantitative variable
distribution
individuals
survey
population
experiment
statistical inference
categorical variable
1. What is meant by a “variable”?
2. Describe the differences between categorical and quantitative data.
3. What is meant by a “distribution”?
4. Explain the key differences between observational studies and experiments. (Consider the
design, implementation, and how the results of each can be interpreted.)
5. Explain the key questions of data analysis from your book.
6. Explain the concept of a “lurking variable”
7.
(a) Describe the explanation of probability as stated in your book.
(b) Then give an example that further demonstrates your understanding of the concept
of probability as long-run behavior.
Displaying Distributions with Graphs (1.1)
Learning Targets:
Describe what is meant by exploratory data analysis
Explain what is meant by the distribution of a variable
Differentiate between categorical and quantitative variables
Construct bar graphs and pie charts for a set of categorical data
Construct a stemplot for a set of quantitative data
Construct a back-to-back stemplot to compare two related distributions
Construct a stemplot using split stems
Construct a histogram for a set of quantitative data, and discuss how changing the class width can
change the impression of the data given by the histogram
Describe the overall pattern of a distribution by its center, shape, and spread
Explain what is meant by the mode of a distribution
Recognize and identify symmetric and skewed distributions
Explain what is meant by an outlier in a stemplot or histogram
Construct and interpret an Ogive from a relative frequency table
Construct a time plot for a set of data collected over time
CASE STUDY:
What does it mean to say that a TV show was ranked number 1? The
Neilsen Media Research Company randomly samples about 5100 households and 13,000
individuals each week. The TV viewing habits of this sample are captured by metering
equipment, and data are sent automatically in the middle of the night to Neilsen. Broadcasters
and companies that want to air commercials on TV use the data on who is watching TV and what
they are watching. The results of this data gathering appear as ratings on a weekly basis.
Here is an alphabetical list of the top 20 prime-time shows for viewers aged 18 to 49 during
the week of November 22-28, 2004:
Show
Network
According to Jim
Amazing Race
Apprentice
Biggest Loser
Boston Legal
CSI
CSI: Miami
CSI: NY
Desperate Housewives
Extreme Makeover: Home Edition
Fear Factor
Law & Order: SVU
Monday Night Football
NFL Monday Showcase
Raymond
Seinfeld
Survivor: Vanuata
Two and a Half Men
60 Minutes
ABC
CBS
NBC
NBC
ABC
CBS
CBS
CBS
ABC
ABC
NBC
NBC
ABC
ABC
CBS
NBC
CBS
CBS
CBS
Which network is winning the ratings battle?
Viewers (millions)
5.5
6.1
7.1
5.4
7.4
10.9
10.5
6.1
16.2
9.7
6.5
7.8
7.8
5.7
8.0
7.6
7.8
8.8
5.4
Individuals :
Variables :
Categorical variables:
Quantitative variables:
Univariate data:
Graphs are used to display data. (There will be a graph question on the AP Exam)
Categorical variables are displayed with one of the following graphs:
Bar graph
Pie chart
Time plot
Quantitative variables are displayed with one of the following graphs:
Dotplot
Stemplot (Stem-and-leaf)
Frequency Distribution
Relative frequency/Cumulative frequency
Histogram
When we describe the graph, we are describing the distribution of the quantitative
variable. Look for an overall pattern (use the terms “center”, “shape”, and “spread”).
Center and spread are discussed in the next section. Shape refers to:
Symmetric
Skewed (right-skewed vs. left-skewed)
Example: The population of the United States is aging, though less rapidly than in
other developed countries. Here is a stemplot of the percents of residents aged 65
and over in the 50 states, according to the 2000 census. The stems are whole
percentages and the leaves are tenths of a percent.
5
7
Florida)
6
7
8
5
9
679
10
6
the
11
02233677
12
0011113445789
13
00012233345568
14
034579
15
36
16
17
6
1. There are 2 outliers (Alaska and
What are the percents for these two
states?
2. Ignoring Alaska and Florida, describe
shape, center, and spread of this
distribution
3. Re-graph this data using a histogram
Enter the data into a list in your calculator (STAT  EDIT  L1)
Sort the list in descending order (STAT  SortD  L1)
Frequency Distribution:
CLASSES
FREQUENCY
RELATIVE FREQ. (%)
RELATIVE
CUMMULATIVE FREQ
Histogram:
Describing Distributions with Numbers (1.2)
Learning Targets
Given a data set, compute the mean and median as measures of center
Explain what is meant by a resistant measure
Given the data set, find the quartiles
Given a data set, find the five-number summary
Use the five-number summary of a data set to construct a boxplot for the data
Compute the interquartile range (IQR) of a data set
Given a data set, use the 1.5(IQR) rule to identify outliers
Given a data set, compute the standard deviation and variance as measures of spread
Identify situations in which the mean is the most appropriate measure of the center and situations in
which the median is the most appropriate measure
MEASURES OF CENTER: Mean, Median, Mode
A small company consists of the owner, a manager, a salesperson, and 2 technicians.
Their annual salaries are listed below:
Staff
owner
manager
salesperson
technician
technician
Salary
$200,000
$50,000
$25,000
$15,000
$15,000
What is the Median? ________________
What is the Mode? ______________
Mean =
sum of values x1  x 2  x 3  ...  xn

# of values
n
 xi
x n
1 x
n
 i
Why is the mean so much higher than the median?
NONRESISTANT measure of center:
RESISTANT measure of center:
(unimodal vs. bimodal)
If data is symmetric, mean and median are the same.
If data is skewed, the mean is farther out in the tail than the median.
MEASURES OF SPREAD: Range, Quartiles, Standard Deviation
1.) Range:
2.) Quartiles:
3.) Standard Deviation:
Your calculator will compute a 5-number summary that includes:
min value
Q1
median
Q3
which you can then graph as a box plot.
max value
Example: Weights of Cowley County Community College Volleyball Players:
131
134
114
188
167
175
180
133
a) Find the 5-number summary for the above data
b) Convert this information to a boxplot
126
130
265
110
c) Check your boxplot with the calculator
d) Perform the outlier test:
Q1 – 1.5(IQR) = low-end boundary
Q3 + 1.5(IQR) = high-end boundary
Example: Two different brands of paint were tested to see how long each would last
before fading. The following data lists the number of months the paint lasted:
Brand A
10
60
50
30
40
20
x = 35
Brand B
35
45
30
35
40
25
x = 35
The means are the same, but Brand B was more consistent…the data was clustered
around the mean more than Brand A.
Brand A
distance from x (deviations)
squared deviations
sum = ________
sum = ________
10
60
50
30
40
20
Use your calculator to calculate the squared deviations for Brand B
Since the sum of the deviations is always zero, we can find the last deviation just by
figuring out what number needs to make the group sum to zero. This means all but the
last number are free to vary, but the last one must make the group add to zero. If
there are n data values, then there are n-1 values that are free to vary. We call this
“n-1 degrees of freedom.” To continue calculating the standard deviation, we divide by
n-1 to find the “average”, which gives us the variance.
Variance for Brand A =
Variance for Brand B =
what we know from these numbers is that
Brand A is more spread out. That’s it,
nothing else.
s = standard deviation for the data =
var iance
Standard deviation for Brand A =
Standard deviation for Brand B =
Standard Deviation can be thought of as the “typical distance the data is from the
mean”.
Chapter 1: Exploring Data
Key Vocabulary:
center
dot plot
Range
shape
histogram
time plot
mean
median
resistant
IQR
minimum
standard deviation
spread
skewed left
stemplot

nonresistant
maximum
variance
frequency
skewed right
split stems
outlier
symmetric
back-to-back stemplot
x
5-number summary
quartiles
Q 1 , Q3
boxplot
modified boxplot
degrees of freedom
1.1 Displaying Distributions With Graphs
1. Explain “roundoff error”
2. When is it useful to use a bar chart?
3. When is it useful to use a pie chart?
4. Define “range”
5. When is it better to use a histogram rather than a dotplot?
6. What is meant by “frequency” in a histogram?
7. Draw examples of a symmetric histogram, skewed right histogram, and skewed left
histogram.
8. Explain a split-stem stemplot.
9. Explain a back-to-back stemplot.
10. Explain a modified boxplot.
11. Explain a parallel boxplot.
1.2 Describing Distributions with Numbers
12. In statistics, what is the most common measurement of center?
13. Explain how to calculate the mean, x
14. Explain why the median is resistant to extreme observations, but the mean is nonresistant.
15. What does standard deviation measure?
16. What is the five-number-summary?
17. What is the relationship between variance and standard deviation?
18. When does standard deviation equal zero?
19. Is standard deviation resistant or nonresistant to extreme observations? Explain.
Write a 20-word summary of the reading assignment that captures the main ideas of
the 1st chapter.
Measures of Relative Standing and Density Curves (2.1)
Learning Targets
Compute the z-score of an observation given the mean and standard deviation
Explain what is meant by a standardized value
Compute the “pth” percentile of an observation
Explain what is meant by a mathematical model
Define a density curve
Explain where the mean and median of a density curve are to be found
Describe the relative position of the mean and median in a symmetric density curve and in a skewed
density curve
Density Curves
What would you say about the distribution of data in this histogram?
Center:
Shape:
Spread:
Since we’ve been estimating values when using a histogram, and we’re really only
interested in the shape (most of the time), we can smooth out the tops of the bars.
Mathematical model :
*
*
*
A density curve could be any shape:
Since the density curve is an approximation, the mean and standard deviation of the
curve might be different from the actual observed values.
x =
 =
s=
 =
Measures of Relative Standing: z-scores
Here are the scores of all 25 students in Mr. Pryor’s statistics class on their first test:
79
77
81
83
80
86
77
90
73
79
83
85
74
83
93
89
78
84
80
82
75
77
67
72
73
Jenny scored an 86. How did she perform relative to her classmates?
We could look at the stemplot.
Notice it is roughly symmetric
with no apparent outliers.
What can we conclude from these displays?
We could look at the Minitab output
of summary statistics for the test scores.
STANDARDIZING -
Z-SCORE:
EXAMPLE: Three landmarks of baseball achievement are Ty Cobb’s batting average of
0.420 in 1911, Ted Williams’s 0.406 in 1941, and George Brett’s 0.390 in 1980. These
batting averages cannot be compared directly because the distribution of major league
batting averages has changed over the years. The distributions are quite symmetric,
except for the outliers such as Cobb, Williams, and Brett. While the mean batting
average has been held roughly constant by rule changes and the balance between
hitting and pitching, the standard deviation has dropped over time. Here are the facts:
Decade
1910s
1940s
1970s
Mean
0.266
0.267
0.261
Std. Dev.
0.0371
0.0326
0.0317
Which player stood out most from his peers?
zCobb =
zWilliams =
zBrett =
Measures of Relative Standing: percentiles
In chapter 1, we defined the pth percentile of a distribution as the value with p percent of the
observations less than or equal to it. If the distribution is normal, we can use z-scores to
calculate percentiles. A z-score of 0 has 50% of the observations below it and 50% of the
observations above it. You can use a table of values to calculate percentiles.
TABLE “A” IS IN YOUR TEXTBOOK, AND A COPY OF IT IS IN YOUR BINDER
A z-score of -2.24 has
0.0125 of the
area/observations below it.
The z-score corresponding to
1.58% of the
area/observations is -2.15
EXAMPLE: The distribution of heights of young women aged 18 to 24 is
approximately normal with mean  = 64.5 inches and standard deviation  = 2.5 inches.
(a) What percent of girls are shorter than 69.5 inches?
(b) What percent of girls are taller than 67 inches?
(c) What percent of girls are between 59.5 in. and 67 in.?
(d) Taylor Swift is approximately 5’11”. How does her height compare to the other
girls?
Normal Distributions (2.2)
Learning Targets:Identify the main properties of the Normal curve as a particular density curve.
List three reasons why Normal distributions are important in statistics
Explain the 68-95-99.7 rule (the empirical rule)
Explain the notation N
Density Curve:
, 
Define the standard Normal distribution
Use a table of values for the standard Normal curve (Table A) to compute the
proportion of observations that are (a) less than a given z-score, (b) greater
than a given z-score, or (c) between two given z-scores.
Use a table of values for the standard Normal curve to find the proportion of
observations in any region given any Normal distribution (i.e. given raw data
rather
than z-scores)
Use a table of values for the standard Normal curve to find a value with a given
proportion of observations above of below it (inverse Normal)
Identify at least two graphical techniques for assessing Normality
Explain what is meant by a Normal probability plot; use it to help assess the
Normality of a given data set
Use technology to perform Normal distribution calculations and to make Normal
probability plots
Normal Curve:
The exact density curve for a particular distribution is described by giving its mean
and its standard deviation. The usual notation is: ________________
To sketch a Normal curve, draw a number line and mark the mean. Put the peak of the curve
above the mean.
Label the number line in intervals of your standard deviation. Always include 3 intervals to the
left and 3 intervals to the right.
“concave down”
“concave up”
Practice: Sketch the distribution N(15.8, 1.2)
All Normal distributions obey the following rule:
_____
_
_____
_
________
_
This rule is sometimes known as the “Empirical Rule”
_________________
_
_____________
_
_____________
_
Example: The distribution of heights of American men aged 20 – 29 is approximately
Normal with mean 70 inches and standard deviation 3 inches.
(a) Draw a Normal curve on which the mean and standard deviation are correctly
located.
(b) What percent of men are taller than 69.5 inches?
(c) Between what heights do the middle 95% of men fall?
(d) What percent of men are shorter than 59.5 inches?
(e) A height of 67 inches corresponds to what percentile of adult American men’s
heights?
Example:
The level of cholesterol in the blood is important because
high cholesterol levels may increase the risk of heart disease. The
distribution of blood cholesterol levels in a large population of people
of the same age and gender is roughly Normal. For 14-year-old boys,
the mean is   170 milligrams of cholesterol per deciliter of blood
(mg/dl) and the standard deviation is
  30 mg/dl. About 1% of
14-year-old boys have cholesterol high enough to require medical
attention. What cholesterol level requires medical attention in
14-year-old boys?
(a) Sketch the distribution and mark the important points on the horizontal axis.
(b 1) Use table A to find the standardized value
(b 2) Use InvNorm to find the standardized value
(c) Un-standardize the variable
(d) State your conclusion
Assessing Normality
There are some methods we can use to determine if a distribution
might be Normally distributed.
I. Construct a histogram or stemplot. Look for symmetry and an approximate bell shape.
II. Construct a Normal Probability Plot
a. Arrange data values from smallest to largest (Use L 1)
b. Find corresponding z-scores for each x-value (Enter into L2)
c. Plot each data point (x-value) against its corresponding z-score. (Scatterplot) If the result is a
fairly straight line, then the data values came from an approximately Normal distribution.
Example: According to USA Today, the 2011 salaries for the Kansas City Royals are as
follows. Assess the Normality of this data.
Kansas City Royals
RK PLAYER
1. Billy Butler
2. Jeff Francoeur
3. Alex Gordon
4. Bruce Chen
5. Jonathan Broxton
6. Luke Hochevar
7. Felipe Paulino
8. Aaron Crow
9. Humberto Quintero
Alcides Escobar
Salary (US$)
8,500,000
6,750,000
6,000,000
4,500,000
4,000,000
3,510,000
1,900,000
1,600,000
1,000,000
1,000,000
RK
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
PLAYER
Chris Getz
Jose Mijares
Brayan Pena
Mitch Maier
Eric Hosmer
Greg Holland
Tim Collins
Luis Mendoza
Mike Moustakas
Kelvin Herrera
Salary (US$)
967,500
925,000
875,000
865,000
502,500
497,150
495,725
488,925
487,250
480,650
Chapter 2: The Normal Distributions
Key Vocabulary:
 mu
Density curve
Inflection point
Normal distribution
Standard Normal
z-scores
 sigma
Empirical rule
distribution
Outcomes
Percentile
Normal Probability plot
Simulation
N
Normal curve
Standardized value
normalcdf
invNorm
2.1
 ,  
Density Curves and Measures of Relative Standing:
1. Explain the two ways to describe the position of Jenny’s score on the first statistics test within the
distribution of test scores.
2. How do we calculate the percent of observations falling within k standard deviations of the mean?
3. List the 4 steps used to explore data from a single quantitative variable.
4. Describe the relationship between the mean and the median of a skewed distribution.
2.2
Normal Distributions:
5. List the three reasons that normal distributions are important in statistics.
6. Give 3 examples of distributions that are often close to normal.
7. How is a standard normal distribution different from a normal distribution?
8. List the steps used in solving problems involving normal distributions.
9. In what situation do we use Table A backwards?
10. Is there a difference between the 80th percentile and the top 80%? Explain
11. Explain the basic idea of a Normal probability plot.
12. How does a Normal probability plot indicate that the data are normal?
13. Write 5 – 6 sentences about your progress in AP Statistics so far.
Scatterplots and Correlation (3.1)
Learning Targets:
Explain the difference between an explanatory variable and a response variable
Given a set of bivariate data, construct a scatterplot
Explain how to recognize an outlier in a scatterplot
Explain what it means for two variables to be positively or negatively associated
Explain how to add categorical variables to a scatterplot
Use a TI83 / TI84 to construct a scatterplot
Define the correlation “r” and describe what it measures
Explain what is meant by the direction, form, and strength of the overall pattern of a scatterplot.
Given a set of bivariate data, use technology to compute the correlation “r”
List the four basic properties of the correlation “r” that you need to know in order to interpret any
correlation
List four other facts about correlation that must be kept in mind when using “r”
Explanatory variables ____________________________________________
Response variables ______________________________________________
EXAMPLE: One effect alcohol has on the body is a drop in body temperature
Explanatory variable: _____________________Response variable: __________________
EXAMPLE: Can we predict a state’s average SAT math score if we know the average SAT
verbal score?
Explanatory variable: ___________________Response variable: _____________________
Note:
_________________________________________________________________
_________________________________________________________________________
Scatterplots show the relationship between two quantitative variables measured on the same
individuals.
We plot the explanatory variable on the x-axis, and plot the response variable on the y-axis.
To interpret scatter plots, look for the overall pattern and for striking deviations from that
pattern.
To describe a pattern look at association, form, and strength:
Association: ______________________________________________________
Form: ___________________________________________________________
Strength:
______________________________________________________
What is the direction of these scatterplots?
Which scatterplot is stronger?
Note:___________________________________________________________
________________________________________________________________________.
The correlation, r, measures the ___________________________________ of linear
associations between quantitative variables.
To calculate r:
Notice that
xi  x
y y
represents the standardized values of x and i
represents the
sx
sy
standardized values of y. This allows us to ___________________________________.
**
r is always a value between _________. Strong correlations produce r values close
to __________. The closer r is to 0, the ____________________ the linear
relationship is.
**
Positive r -values indicate _______________; negative r -values indicate a
_________________________.
**
r has no ___________________________.
**
When calculating r. it doesn’t matter which variable is _______________ and
which one is ________________.
EXAMPLE:
A food industry group asked 3368 people to guess the number of calories in
each of several common foods. Here is a table of the average of their guesses and the correct
number of calories:
Food
Guessed calories
Correct calories
8 oz whole milk
196
159
5 oz spaghetti with tomato sauce
394
163
5 oz macaroni with cheese
350
269
one slice wheat bread
117
61
one slice white bread
136
76
2 oz candy bar
364
260
saltine cracker
74
12
medium-sized apple
107
80
medium-sized potato
160
88
cream-filled snack cake
419
160
We think that how many calories a food actually has helps explain people’s guesses of how many
calories it has. With this in mind, make a scatterplot of these data and describe the
relationship. Calculate the correlation r by hand using the lists on your calculator. Here are
the keystrokes:
Let:
L1 = x-values
(type them in)
L2 = y-values
L3 =
xi  x
sx
(find x and sx using 1-var stats on L1 , L2)
L4 =
yi  y
sy
(find y and sy using 1-var stats on L2)
 xi  x 

 sx 
L5 = 

 yi  y 


s
y


(multiply L3 and L4)
Find the sum of L5 using: List (2nd Stat)  Math  5: Sum
(type them in)
L1
L2
159
196
153
394
269
350
61
117
76
136
260
364
12
74
80
107
88
160
160
419
L3
Sum of L5 = ______________________
1
 sum L5  = ___________________ = r
n 1
L4
L5
Least Squares Regression (3.2)
Learning Targets:
Explain what is meant by a regression line
Given a regression equation, interpret the slope and y-intercept in context
Explain what is meant by extrapolation
Explain why the regression line is called the “least squares regression line” (LSRL)
Explain how the coefficients of the regression equation y = a + bx can be found given r, sx, sy, and (x , y)
Given a bivariate data set, use technology to construct a least-squares regression line
Define a residual
Given a bivariate data set, use technology to construct a residual plot for a linear regression
List two things to consider about a residual plot when checking to see if a straight line is a good model for a
bivariate data set.
Explain what is meant by the standard deviation of the residuals
Define the coefficient of determination, r, and explain how it is used in determining how well a linear model
fits its bivariate set of data.
List and explain four important facts about least-squares regression
Regression
Since linear relationships between quantitative variables are quite common, it is useful to
summarize overall patterns by drawing a line on a scatterplot. This line is called a regression
line.
A REGRESSION LINE (or line of best fit) ______________________________________
______________________________________________________________________

Regression lines allow us to make predictions
Regression lines model ____________ (like density curves in ch.2) to give us a concise
mathematical description of the relationship between the variables.
There are many methods for drawing a line of fit. We need a method of drawing a line of fit
that does not depend on guessing where the line should be. In the field of statistics, the
model we use is called the Least Squares Regression Line (LSRL).
Since we are using the line to predict
y from x, a good line of fit will make
the vertical distances of the points
from the line as small as possible.
The error is calculated by subtracting the predicted
y-value from the actual y-value:
error = observed (actual) response - predicted
response
Residual = Actual – Predicted

(R A P)
The vertical error is called the residual
The least-squares regression line of y on x is the line that makes the sum of the
squared vertical distances of the data points from the line as small as possible

If our regression line is “perfect”, the points will have the same distance above the
line as distance below.

So it’s possible to get a total of zero error when we add all of the residuals.

For this reason, when discussing residuals, we square the errors to eliminate the
negatives (this strategy should sound familiar to you…) and then “undo” the
squaring later.
ŷ
is read “y hat” to emphasize that this is a PREDICTED response for any x.
b is the PREDICTED slope.
Note: When explaining anything about regression lines or regression slopes, be sure
to state that these are PREDICTED values
Example: When purchasing a car, there is a relationship between the price of the car and
the age of the car. Find the LSRL, the value of r, and the residuals using the calculator .
Price of a Saturn SL1 from 10
classified car ads in a
newspaper
Age (years)
Asking Price ($)
1.0
1.0
2.0
2.0
3.0
4.0
5.0
5.0
6.0
6.0
11875
10995
8500
9995
8995
6995
4450
5500
4400
4800
Solution:
1st
Determine which variable is
explanatory and which is response.
x = _____________________
y = _____________________
2nd
To find r: Under “CATALOG”
find “DIAGNOSTIC ON”, then
press ENTER
3rd
then use
L2, Y1
Enter data into lists L1 and L2
STAT  CALC  4: a+bx, L1,
(this tells the calculator to use list 1
and list 2 in finding the regression
equation, and then place the equation into
Y= screen)
y = ________________________________
4th Calculate the residuals for this data:
r = ________________________
Price of a Saturn SL1 from 10 classified car ads in a newspaper
L1
L2
1.0
11875
1.0
10995
2.0
8500
2.0
9995
3.0
8995
4.0
6995
5.0
4450
5.0
5500
ŷ
L3 = predicted values
Residual
L4 = actual - predicted
6.0
4400
6.0
4800
Residual plots should have no clear pattern, with points uniformly scattered above and below
the line. Make a residual plot in the STAT PLOT using L1 for the X List and L4 for the Y List.
When sketching it on your paper, always be sure to label the axes!
RESIDUALS
X – VALUES
Correlation and Regression Wisdom (3.3)
Learning Targets:
Recall the three limitations on the use of correlation and regression
Explain what is meant by an outlier in bivariate data
Explain what is meant by an influential observation and how it relates to
regression
Given a scatterplot in a regression setting, identify outliers and influential
observations
Define a lurking variable
Give an example of what it means to say “association does not imply causation”
Explain how correlations based on averages differ from correlations based on
individuals
Outliers and Influential Points:
Three limitations on the use of correlation and regression:
1. ______________________________________________________________
2. ______________________________________________________________
3. ______________________________________________________________.
Example:
Does the age at which a child begins to talk predict a later score of mental
ability? A study of the development of young children recorded the age in months at which
each of 21 children spoke their first word and their Gesell Adaptive Score, the result of an
aptitude test taken much later. The results appear in the scatter and residual plots below.
Children 18 and 19 are outside the cluster of the rest of the data. Child 19 is an outlier in
the y-direction. Child 18 is an outlier in the x-direction. It has a strong influence on the
position of the regression line. The graph below adds a 2nd regression line calculated after
leaving out child 18.
What happens to the LSRL?
This applet will show you how the LSRL changes with the addition of outliers and
influential points.
http://statweb.calpoly.edu/chance/applets/LRApplet.html
In the regression setting, not all outliers are influential:
The LSRL is most likely to be influenced by outliers in the x-direction.
Influential points often have small residuals, because they pull the regression line toward
themselves, causing other points’ residuals to increase.
If you suspect an influential point, find the equation of the regression both with and without
the point in question. If the line moves more than a small amount when the point is deleted,
the point is influential.
Lurking Variables:
Another caution about regression and correlation is the effect of a third variable. This third
variable may influence the interpretation of relationships between the explanatory and
response variables.
EXAMPLE: A study was once completed on the relationship between ice cream sales and
snake bite treatments in a local area hospital. The study found a strong positive correlation
between these two variables. Why do you think that is?
Do left-handers die early?
A study of 1000 deaths in California found that left-handed people died at an average age of
66 years old, while right-handed people died at an average age of 75 years old. Should lefthanders fear an early death?
Coefficient of Determination:
When we find the equation for the LSRL we find a value for “r” (correlation coefficient). We
are also given a vale for “r2”, which is called the coefficient of determination.
r2 tells us the fraction of the variance of one variable that is
explained by the LSRL on the other variable.
Basically, it tells us how well the LSRL does at predicting values of the response variable, y.
EXAMPLE:
A study of class attendance and grades among first-year students at a state
university showed that, in general, students who attended a higher percent of their classes
earned higher grades. The coefficient of determination, r 2 = 0.16. Class attendance explained
16% of the variation in grade index among the students.
What else could explain the variation?
What is the numerical value of the correlation between percent of classes attended and grade
index?
Chapter 3: Examining Relationships
Key Vocabulary:
response variable
explanatory variable
independent variable
dependent variable
“y – hat”
negative association
r-squared
r-value
regression line
mathematical model
least-squares regression line
positive association
SSE
correlation
residuals
residual plot
influential observation
scatterplot
SSM
linear
coefficient of determination
3.1 Scatterplots and Correlation:
1. What is the difference between a response variable and an explanatory variable?
2. How are response and explanatory variables related to dependent and independent
variables?
3. When is it appropriate to use a scatterplot to display data?
4. Explain the difference between a positive association and a negative association.
5. Explain what it means to “describe the overall pattern of a scatterplot using direction,
form, and strength”.
6. How do you add categorical variables in a scatterplot?
7. List the 4 cautions about correlation.
3.2 Least-Squares Regression:
8. In what ways is a regression line a mathematical model?
9. What is extrapolation?
10. What is a least-squares regression line?
11. Define residual:
12. If a least-squares regression line (LSRL) fits the data well, what 2 characteristics should
the residual plot exhibit?
13. What numerical quantity tells us how well the LSRL will do at predicting?
14. Explain the idea of r-squared.
3.3 Correlation and Regression Wisdom:
15. What three cautions are reviewed in this section?
16. Explain the difference between an influential point and an outlier.
17. Explain how a lurking variable can make correlation or regression misleading.
18. What are the three new cautions of this section?
Relationships between Categorical Variables (4.2)
Learning Targets:
Explain what is meant by a “two-way table”
Explain what is meant by “marginal distributions” in a two-way table
Describe how changing “counts” to “percents” is helpful
Explain what is meant by a “conditional distribution”
Define “Simpson’s paradox” and give an example of it
Two-way Tables:
We use a two-way table to display relationships between two or more categorical variables.
Some variables are categorical by nature: gender, race, occupation, etc. Other variables
become categorical by grouping quantitative variables into classes.
A two-way table has a row variable and a column variable. The entries in the table can be
counts or percents.
College students by gender and age group, 2003 (thousands of persons)
Age group
Female
Male
Total
15-17 years
18-24 years
25-34 years
35+ years
89
5668
1904
1660
61
4697
1589
970
150
10,365
3,494
2,630
Total
9321
7317
16,639
Identify the row variable and the column variable:
Marginal Distributions:
The distribution of the categorical variables in the table above says how often each outcome
occurred. The “total” column contains the total for each of the rows.
How many college students were ages 18 to 24? __________________
How many college students were ages 15 to 17? ___________________
The “total” row contains the total for each column. How many female college students were
there in 2003?
____________________
Since the totals occur in the margins of the table, they are called marginal distributions.
College students by sex and age group, 2003 (thousands of persons)
Age group
Female
Male
Total
15-17 years
18-24 years
25-34 years
35+ years
89
5668
1904
1660
61
4697
1589
970
150
10,365
3,494
2,630
total
9321
7317
16,639
The marginal distributions from the above table do not tell us how the two variables are
related. The relationship is in the body of the table. To describe relationships among
categorical variables, calculate appropriate percents from the given counts. Counts are often
hard to compare, but percents can tell a lot about the relationship.
What percent of the 15-17 age group is female? __________________________
What percent of the male students were of age 18-24? ______________________
What percent of college students were females between the ages of 25-34? _______
Each of the above answers is called a conditional distribution, because to find the value, we
used a table entry from a given condition.
Do the female and male totals agree with the overall total? __________
We sometimes encounter ROUNDOFF ERROR in two-way tables. The entries above were
rounded to the nearest thousand, so when adding the row totals and the column totals, we
sometimes find a very small discrepancy. As long as you understand why the error is there, it
shouldn’t affect your conclusions.
Simpson’s Paradox:
Lurking variables can also affect categorical variables. They can change or even reverse
relationships.
EXAMPLE:
Accident victims are sometimes taken by helicopter from the accident scene to
a hospital. Helicopters save time, but do they save lives? Below is a comparison between the
counts of accident victims who die with helicopter evacuation and with the usual transport to a
hospital by road.
Helicopter
64
136
200
Victim died
Victim survived
Total
Road
260
840
1100
What percent of helicopter patients died? ___________________
What percent of road transported patients died? _____________________
What do these percents suggest? _______________________________________
There is a lurking variable in this example. What is it? ________________________
Here is the same data broken down differently:
Serious Accidents
Victim died
Victim survived
Total
Helicopter
48
52
100
Less Serious Accidents
Road
60
40
100
Victim died
Victim survived
Total
Helicopter
16
84
100
Check the conditional distributions for death of patients:
Helicopter(serious)/died
Helicopter(less serious)/died
Road(serious)/died
road(less serious)/died
Do these percents suggest the same conclusion as above?
Road
200
800
1000
Establishing Causation (4.3)
Learning Targets:
Identify the three ways in which the association between two variables can
be explained
Explain what process provides the best evidence for causation
Define what is meant by a common response
Defined what it means to say that two variables are confounded
Discuss why establishing a cause-and-effect relationship through
experimentation is not always possible
Explain what it means to say that a lack of evidence for a cause-and-effect
relationship does not necessarily mean that there is no cause-and-effect
relationship
List 5 criteria for establishing causation when you cannot conduct a controlled
experiment
Explaining Association:
When we look at relationships between variables, often we hope to show that changes in the
explanatory variable CAUSE changes in the response variable.
Remember, a strong association between two variables is not enough to determine a cause and
effect relationship.
There are three ways we can explain the association between two variables:
1. CAUSATION: There is a direct cause - effect relationship between two variables.
2. COMMON RESPONSE: The observed association between two variables is explained by
a lurking variable. BOTH variables change due to the lurking variable.
3. CONFOUNDING: Either explanatory variables or lurking variables have mixing
influences and cannot be distinguished from each other.
Reminder: A lurking variable is a variable that is not an explanatory or response variable that
may still influence the relationships among those variables.
It is important to note that even when direct causation is present, very seldom does this
completely explain an association between variables.
EXAMPLE:

SAT scores
Why is a student’s SAT Math score positively associated with SAT Verbal score?
How do we establish causation?
Carefully designed experiments are the best way, but are not always practical, ethical, or even
possible.
What are the criteria for establishing causation when we can’t do an experiment?
Look for the following attributes:
1.
The association is strong.
2.
a.
The association is consistent.
Consistency reduces the chances that a lurking variable is present.
3.
Larger values of the response variable are associated with stronger responses.
The alleged cause precedes the effect in time.
The alleged cause is plausible.
4.
5.
EXAMPLE A: Doctors had long observed that most lung cancer patients were smokers.
Comparison of smokers and similar non-smokers showed a very strong association between
smoking and death from lung cancer.
Could the association be due to common response? Might there be, for example, a genetic
factor that predisposes people both to nicotine addiction and to lung cancer? If so, smoking
and lung cancer would be positively associated even if smoking had no direct effect on the
lungs. (Genetics would be a lurking variable affecting the smoking and the lung cancer)
Or perhaps confounding was to blame. It might be that smokers live unhealthy lives in other
ways such as an unhealthy diet, too much alcohol, or lack of exercise, which could cause the
cancer. In this case, another habit might be influencing the lung cancer that has nothing to do
with smoking.
We can’t design an experiment where we force some patients to smoke and some not to smoke this would be unethical. Yet, medical authorities do not hesitate to say that smoking causes
lung cancer. So, how are they so sure? Let’s look at the five criteria:
1. Strong association: The association between smoking and lung cancer is very strong.
2. Consistent association: Many countries studied the association and found similar results.
3. Larger values of response are related to stronger responses: People who smoke more
cigarettes per day, or who smoke over a longer period develop lung cancer at a higher rate
than people who stop smoking.
4. Cause precedes effect: Lung cancer develops after years of smoking. When smoking
became more common, lung cancer cases increased about 30 years after.
5. Plausible: Experiments on animals show that tar from cigarette smoke causes cancer in
the animal.
Each of the following examples shows that causation is not a simple idea:
CAUTION
Even well-established causal relations may not generalize to other settings.
EXAMPLE B:
Experiments have conclusively shown that large amounts of saccharin in the
diet cause bladder cancer in rats. Should we avoid saccharin as a sugar substitute?
*
*
CAUTION
Even when direct causation is present, it is rarely a complete explanation of
an association between the two variables.
EXAMPLE C:
A study of Mexican-American girls aged 9 to 12 years old recorded body
mass index (BMI), a measure of weight relative to height, for both the girls and their
mothers. The study also measured hours of television viewing, minutes of physical activity,
and intake of several kinds of food. The strongest correlation (r = 0.506) was between the
BMI of daughters and the BMI of mothers. Yet, the mother’s BMIs explain only 25% (r2) of
the variance among the daughters’ BMIs. What else could explain the variance among the
daughters’ BMI?
*
*
Chapter 4: More about Relationships between Two Variables
Key Vocabulary:
Exponential function
Power function
Linear growth
Exponential growth
Extrapolation
Power law model
Simpson’s paradox
lurking variables
causation
confounding
common response
marginal distributions
conditional distributions
4.2 Relationships between Categorical Variables
1. What is used to analyze categorical variables?
2. What is a two-way table?
3. Describe roundoff error.
4. Why are percents used to describe the relationship between categorical variables?
5. Explain the difference between marginal distributions and conditional distributions.
6. Explain Simpson’s Paradox.
4.3 Establishing Causation
1. Define causation and give an example.
2. Explain confounding and give an example.
3. Explain common response and give an example.
4. Give an example of a potential cause-and-effect situation that cannot be verified by the use
of an experiment.
5. Comment on your performance in the class so far. Is there anything you can do to be more
successful? Is there anything your instructor can do to help you be more successful?
Designing Samples (5.1)
Learning Targets:
Define: Population, Sample, Biased, Simple Random Sample (SRS), Systematic Random
Sampling, Probability Sample, Cluster Sample, Undercoverage, Nonresponse
Explain: Voluntary Response Sample, Sampling vs. Census, Convenience Sampling,
Observational study vs. Experiment
Give examples of: Voluntary Response Sample, Response Bias
Determine: 4 steps involved in choosing an SRS, Strata of interest, Major advantage of large random
samples
Introduction:
The chapters we have studied taught us how to analyze data by looking at patterns and
departures from patterns. Now, we will see how to produce the data. Data can be gathered
from observational studies and experiments.
Observational Study: _____________________________________________
_________________________________________________________________
Experiment: _____________________________________________________
_________________________________________________________________
EXAMPLES:
Which technique (observational study or experiment) for gathering data do
you think was used in the following studies?
1. The Colorado Division of Wildlife netted and released 774 fish at Quincy Reservoir.
There were about 219 perch, 315 blue gill, 83 pike, and 157 rainbow trout.
2. The Colorado Division of Wildlife caught 41 bighorn sheep on Mt. Evans and gave each
one an injection to prevent heartworm. A year later, 38 of these sheep did not have
heartworm, while the other 3 did.
3. The Colorado Division of Wildlife imposed special fishing regulations on the Deckers
section of the South Platte River. All trout fewer than 15 inches had to be released. A
study of trout before regulation length and after regulation
lengths showed that the
average length of a trout increased by 4.2 inches after the regulation went into effect.
4. An ecology class used binoculars to watch 23 turtles at Lowell Ponds. It was found that
18 were box turtles, and 5 were snapping turtles
We usually want to gather information about a large group of individuals. We use certain
symbols when discussing the data.
Population: ________________________________________________________
Population mean:
Population standard deviation::
Sample: __________________________________________________________
Sample mean:
Sample standard deviation:
EXAMPLES:
For each of the following sampling situations, identify the population and the
sample as exactly as possible.
1. Each week, the Gallup Poll questions a sample of about 1500 adult U.S. residents to
determine national opinion on a wide variety of issues.
2. The 2000 Census tried to gather basic information from every household in the United
States. A “long form” requesting much additional information was sent to a sample of about
17% of households.
3. A machinery manufacturer purchases voltage regulators from a supplier. There are
reports that variation in the output voltage of the regulators is affecting the performance
of the finished products. To assess the quality of the supplier’s production, the
manufacturer sends a sample of 5 regulators from the last shipment to a laboratory for
study.
There are two distinct ways of gathering data:
Sampling: _______________________________________________________
Census: _________________________________________________________
Methods of Sampling:
Voluntary Response Sampling: _________________________________________
_________________________________________________________________
Convenience Sampling: _______________________________________________
_________________________________________________________________
Simple Random Samples: (SRS) _______________________________________
_________________________________________________________________
Systematic Random Sampling: _________________________________________
_________________________________________________________________
Stratified Random Sample: ___________________________________________
_________________________________________________________________
_________________________________________________________________
Cluster Sampling: __________________________________________________
_________________________________________________________________
_________________________________________________________________
The best sampling method is to uses an SRS, since it reduces bias the best. The easiest SRS
is to put all names in a hat and draw names. Another method of selecting an SRS is the use of
a Random Digit Table (Table B in the back of the book). We will also use the random digit
table in chapter 6 to conduct a simulation.
EXAMPLE: Suppose we are interested in the number of students in all of the Blue Valley
high schools who have ever spent more than a week in another country. In the interest of
time, we choose to select a sample.
1. How could we choose our sample using the Voluntary Response Sampling method?
2. How could we choose our sample using the Convenience Sampling method?
3. How could we choose our sample using the Systematic Random Sampling method?
4. How could we choose our sample using the Cluster Sampling method?
5. How could we choose our sample using the Simple Random Sampling (SRS) method?
Cautions about Sample Surveys
Random selection eliminates bias in the choice of a sample. When the population consists of
human beings, however, it takes more than a good sampling method to get accurate
information.
Undercoverage: ____________________________________________________
There is rarely a list of every possible participant available in a population.
household sample survey –
telephone opinion polls –
Nonresponse: ______________________________________________________
Response Bias: ____________________________________________________
Inference about the Population
Using chance to select a sample eliminates bias in the actual selection of the sample, but it is
unlikely that results from a sample are exactly the same as for the entire population. Later in
the course, we will discuss how close we need to be in order to generalize our sample results to
the entire population.
If we select two sample at random from the same population, we will draw different
individuals, so the sample results will almost certainly differ somewhat. We can improve our
results by knowing that larger random samples give more accurate results than smaller
samples.
Designing Experiments (5.2)
Learning Targets:
Define experimental units, subjects, treatment, factor, level, placebo effect, replication,
randomization, completely randomized design, block, matched pairs design, and double blind
Given a number of factors and the number of levels for each factor, determine the number of
treatments
Explain the major advantage of an experiment over an observational study
Explain the difference between control and a control group, and explain the purpose of each
Use the random-digit table to assign individuals to a treatment group or a control group
List the 3 main principles of experimental design
Explain the phrase “statistically significant”
Generate an outline, followed by an explanation, of a completely randomized design for an experiment
Experimental Units, Subjects, and Treatments
A study is an experiment when we actually do something to individuals in order to observe the
response. What we do is called the treatment.
The explanatory variables in an experiment are often called ___________________.
Many experiments study the joint effects of several ________________. In such an
experiment, each treatment is formed by combining a specific value of each of the
_____________, called a ____________________.
EXAMPLE:
An experiment investigating the effects of repeated exposure to an
advertising message used undergraduate students as subjects. All subjects viewed a 40minute television program that included ads for a digital camera. Some subjects saw a 30second commercial, others saw a 90-second version. The same commercial was shown 1, 3, or 5
times during the program.
This experiment has 2 factors: _________________________________________
The first factor has 2 levels: ___________________________________ and the
second factor has 3 levels:__________________________________.
One level of each factor will create ___________treatments:
Basic Principles of Statistical Design:
The FIRST basic principle of statistical design of experiments is Control:
Some laboratory experiments can get away with having a design as simple as:
Treatment

Observe Results
For example, we may place a heavy object on a support (treatment) and measure how
much it bends (observation). A controlled environment is protection from lurking
variables. When experiments are done in the field or with living subjects, we cannot
determine if the response is from the treatment or a _______________________.
EXAMPLE:
“Gastric freezing” is a clever treatment for ulcers in the upper intestine. The
patient swallows a deflated balloon with tubes attached, then a refrigerated liquid is pumped
through the balloon for an hour. The idea is that cooling the stomach will reduce its
production of acid and so relieve ulcers. An experiment reported in the Journal of the
American Medical Association showed that gastric freezing did reduce acid production and
relieve ulcer pain. The treatment was safe and easy, and was widely used for several years.
The design of the experiment was:
Gastric freezing
Observe pain relief

The experiment was poorly designed. The patients’ response may have been due to the placebo
effect. A placebo is a dummy treatment. Many patients respond favorably to any treatment,
even a placebo. This may be due to trust in the doctor and expectations of a cure, or simply to
the fact that medical conditions often improve without treatment. A later experiment divided
ulcer patients into two groups. One group was treated by gastric freezing as before. The
other group received placebo treatment in which the liquid in the balloon was at body
temperature rather than freezing. The results: 34% of the patients in the treatment group
improved, but so did 38% of the patients in the placebo group. This and other properly
designed experiments showed that gastric freezing was no better than a placebo, and its use
was abandoned.
Explain the situation of “confounding” in the previous example:
We can defeat confounding by
__________________________________________. The placebo effect and
other lurking variables now operate on both groups. The only difference between the
groups is ________________________________________.
A control group is
___________________________________________________.
Caution: Don’t confuse the terms “control” and “control group” (see page 357)
The SECOND basic principle of statistical design of experiments is
Replication.
If the 2nd gastric-freezing experiment was performed on one person in each group, would we
be able to conclude the ineffectiveness of gastric-freezing?
Replication (several subjects in each treatment group) promotes similar responses within each
treatment group, but different from other treatment groups.
The THIRD basic principle of statistical design of experiments is
Randomization.
Comparison of the effects of several treatments is valid only when all treatments are applied
to similar groups of experimental units. What would have happened if the control group in the
second gastric-freezing experiment consisted of patients whose ulcers were 10 times worse
than the ulcers of the patients in the treatment group?
We must rely on chance to assign subjects to the groups. This way, subjects with similar
characteristics are less likely to be grouped together and confounding is minimized.
EXAMPLE:
Does talking on a hands-free cell phone distract drivers? Undergraduate
students “drove” in a high-fidelity driving simulator equipped with a hands-free cell phone.
The car ahead brakes: how quickly does the subject respond? Twenty students (control
group) simply drove. Another 20 (the experimental group) talked on the cell phone while
driving.
How many factors does this experiment have? _________ How many levels? ________
The researchers needed to divide the 40 student subjects into two groups of 20. One
completely unbiased method is to put the names of the 40 students in a hat, mix them up, and
draw 20. These students would make up the control group and the remaining 20 make up the
experimental group (or vice-versa).
The logic behind the randomized comparative design is as follows:

Randomization produces two groups of subjects that we expect to be similar in
all respects before the treatments are applied.

Comparative design helps ensure that influences other than the cell phone
operate equally on both groups.

Therefore, differences in average brake reaction time must be due either to
talking on the cell phone or to the play of chance in the random assignment of
subjects to the two groups
When describing your experimental design, it is usually helpful to begin with an outline or
diagram of your experiment. Here is the design of the cell phone experiment:
We would then follow this design with an explanation, in paragraph form, of how we choose to
randomly allocate, implement treatments, and do the comparisons. The number of treatments
and method of allocation can make our diagram more complicated. Be sure to include
everything you are doing to minimize confounding in your experiment.
EXAMPLE:
Does regularly taking aspirin help protect people against heart attacks? The
Physicians’ Health Study was a medical experiment that helped answer this question. In fact,
the Physicians’ Health Study looked at the effects of two drugs: aspirin and beta-carotene.
The body converts beta-carotene into vitamin A, which may help prevent some forms of
cancer. The subjects were 21,996 male physicians. There were two factors, each having two
levels: aspirin (yes or no) and beta-carotene (yes or no). These factors and levels combined to
form four treatments. One-fourth of the subjects were assigned to each of these
treatments. On odd-numbered days, the subjects took either a white tablet that contained
aspirin or a dummy pill that looked and tasted like the aspirin but had no active ingredient.
This tablet is called a ______________________. On even-numbered days they took either
a blue capsule containing beta-carotene or a dummy pill mimicking the beta-carotene.
Draw a diagram of this experiment:
EXAMPLE (continued) There were several response variables – the study looked for heart
attacks, several kinds of cancer, and other medical outcomes. After several years, it did not
appear that beta-carotene had any effect, but approximately 24% of the placebo group had
suffered a heart attack, while almost 14% of the aspirin group suffered heart attacks. This
difference is large enough to support the claim that taking aspirin does reduce heart
attacks. We can conclude causation since the three basic principles of experimental design
were followed.
We hope to see a large enough difference in the responses that it is unlikely to have happened
by chance. We will discuss the requirements of difference in responses during 2nd semester
and will be making decisions on whether or not the results are statistically significant.
Results are statistically significant if ___________________________________
_________________________________________________________________
Other Randomized Comparative Experiments:
There are different designs that we can base our experiments on, depending on the
circumstances.
BLOCK DESIGN
A block is a group of experimental units or subjects that are known to be
_______________ before the experiment begins. The experimenter believes that
this particular characteristic plays an important role in the outcome. In a block
design, the random assignment to treatments is carried out separately within each
block. Blocks are a form of control, since the characteristic to be blocked may act as
a lurking variable.
EXAMPLE: The progress of a certain type of cancer differs in men and women.
A clinical
experiment will compare three therapies for this type of cancer. How can we control the
difference in progression between men and women?
A diagram for this design would look like this:
MATCHED PAIRS DESIGN
A matched pairs design compares just two treatments, but the subjects are matched
in pairs to avoid lurking variables, or the same subjects are used for each treatment.
EXAMPLE:
In the hands-free cell phone experiment, it is possible that everyone in the no
phone group were good drivers, while everyone in the cell phone group was a bad driver. We
can address this issue by creating a matched pairs experiment. Each subject would drive with
the cell phone and without the cell phone. The randomization comes in when deciding which 20
will drive non-phone first and with-phone second. The remaining 20 will drive with-phone first
and non-phone second.
A diagram for this design would look like this:
BLIND / DOUBLE-BLIND
Blindness refers to the knowledge of which treatment group a subject is in. Single
blind studies do not inform the subject of which treatment he/she is receiving. Double
blind studies keep the treatment group a secret from an evaluator as well. Why would
this be necessary?
Lack of Realism
One very serious weakness of experiments is lack of realism. If the situation does not
realistically duplicate the conditions we want, we will not be able to come to a
conclusion. Using a placebo for something with a familiar taste to the subjects can tip
them off that it’s a placebo if it lacks the familiar taste. Just knowing they are
participating in an experiment may affect the responses of your subjects. And the
environments in different laboratories can alter the results when trying to achieve
replication.
Caution – Statistical analysis of an experiment cannot tell us how far the results
will generalize to other settings.
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
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
C. Method Number Three: Stratified Sample
Consider the field as grouped in vertical columns (called strata). Using your calculator or
a random number table, randomly choose one plot from each vertical column and mark
these plots on the grid.
A B C
D
E F G H
I
J
1
2
3
4
5
6
7
8
9
10
D. Method Number Four: Stratified Sample
Consider the field as grouped in horizontal rows (also called strata). Using your
calculator or a random number table, randomly choose one plot from each horizontal
row and mark these plots on the grid.
A 1 2 3 4
B
C
D
E
F
G
H
I
J
5
6
7
8
9 10
Observations:
1) Compare the class boxplots of the sample means obtained from the SRS and the two methods
of stratified sampling.
2) Based on the results of both activities, under what conditions is it more useful to use
stratified sampling?
3) Based on the results of both activities, under what conditions is it more useful to use a simple
random sample?
Chapter 5: Producing Data
Key Vocabulary:
Voluntary response
Sample
Confounded
Population
Design
Convenience sampling
Statistically significant
biased
response bias
simple random sample
systematic random sample
stratified random sample observational study
strata
experimental units
undercoverage
subjects
nonresponse
treatment
double-blind
block design
factor
level
placebo effect
control group
randomization
completely randomized experiment
matched-pairs design
5.1 – Designing Samples
1. Describe the relationship between sample, population, sampling, and a census.
2. Discuss two sampling methods that may not provide reliable results. Give an example
for each.
3. Discuss the sampling methods mentioned in your book that do produce reliable
results (assuming they are designed and conducted appropriately). Give an example for
each.
4. What is bias?
5. Why is a simple random sample usually the best sampling method?
5.2 – Designing Experiments
6. Describe the difference between “control” and “control group”.
7. Summarize the basic principles of experimental design: control, replication, and
randomization. What is the purpose or goal of designing experiments using these
principles?
8. What does statistically significant mean?
9. Describe the experimental concept of “blinding” and “double-blinding.” Why is this
necessary?
Simulation (6.1)
Learning Targets: List 3 methods that can be used to calculate or estimate the chances of an event
Introduction:
occurring
Define “simulation”
List the 5 steps involved in a simulation
Explain what is meant by “independent trials”
Use a random-digit table to carry out a simulation
Given a probability problem, conduct a simulation in order to estimate the probability
desired
Use a calculator to conduct a simulation of a probability problem
Toss a coin 10 times. How could we determine the likelihood of a run of 3 or more consecutive
heads or consecutive tails?
An airline knows from past experience that a certain percent of customers who have
purchased tickets will not show up to board the airplane. If the airline overbooks a particular
flight, what are the chances that the airline will encounter more ticketed passengers than they
have seats for?
There are 3 methods we can use to answer question like these involving chance.
1. Observe the random phenomenon many times and calculate the frequency of the results.
2. Develop a probability model to calculate a theoretical answer.
3. Simulate the random phenomenon by repeating a procedure and calculating the results
Option 1 is not the best choice because ___________________________________
Option 2 may not be feasible because ____________________________________
Option 3 is our best choice because ______________________________________
Simulation:
A simulation is when you imitate an event for learning purposes. When we simulate an
experiment, we will be looking for a probability to use for predicting future outcomes.
ACTIVITY A:
Simulating a coin toss APPS  ProbSim

1 (toss coin)
We would expect 50% of the tosses to be heads and 50% to be tails. Does this happen?
Simulate 80 tosses and record the number of heads tossed. The more simulations we do, the
more our data will resemble the true probability.
CLASS RESULTS
Number of “heads” tossed
F
R
E
Q
U
E
N
C
Y
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
NUMBER OF “HEADS” TOSSED
Simulation Steps:
Step 1: State the problem or describe the random phenomenon
“We are investigating …”
Step 2: State the assumptions
“We are assuming …”
Step 3: Assign digits to represent the outcomes
“Let digits 0, 1, 2 represent … while digits 3 – 9 represent …”
Step 4: Simulate many repetitions
Show the result of each trial
Step 5: State your conclusions: It appears the probability of ____________ is
approximately ________”
l
l
ACTIVITY B:
Suppose we were interested in the likeliness of a “run” of 3 heads or 3 tails
out of 10 tosses of a coin. We would have to keep track of this manually if we used the
probability simulator. But since the likeliness of heads = likeliness of tails and on the
random-digit table, the likeliness of a 1 = likeliness of a 2 = likeliness of a 3 (etc.) we can
use the random-digit table to perform the simulation.
STEP 1: We are investigating __________________________________________
STEP 2: We are assuming _____________________________________________
We could assign ODD digits to heads and EVEN digits to tails, or vice-versa. Or we could
assign digits 0 – 4 to heads and digits 5 – 9 to tails. There are many ways we can make the
assignment of the digits AS LONG AS THE PROBABILITIES MATCH UP.
STEP 3: Let digits _______ represent _______________ while digits _________
represent _____________________
STEP 4: Begin with line 101 and find the likeliness of a run of 3 heads or 3 tails in a total of
10 tosses:
LINE 101
19223
95034
05756
TRIAL 1
28713
96409
12531
TRIAL 3
TRIAL 2
From these 3 trials, what is the probability? ___________ Is this reasonable? _____
Continue with line 101 (wrapping around to lines 102, 103, 104, and so on) and perform 22 more
trials, for a total of 25 trials:
42544
82853
73676
47150
99400
01927
27754
42648
82425
36290
45467
71709
77558
00095
32863
29485
82226
90056
52711
38889
93074
60227
40011
85848
48767
52573
95592
94007
69971
91481
60779
53791
17297
59335
68417
35013
15529
72765
85089
57067
50211
47487
82739
57890
How many of the 25 trials had a run of 3? ________
What is the likeliness of getting a run of 3 heads or 3 tails in 10 tosses? __________
STEP 5: It appears _____________________________________________
ACTIVITY C:
Suppose that 80% of a university’s students favor abolishing evening
exams. You ask 10 students chosen at random. What is the likelihood that all 10 favor
abolishing exams?
STEP 1:
STEP 2:
STEP 3: Assign digits _________ to represent “yes” and digits ___________ to
represent “no”
STEP 4: Simulate 25 repetitions (of 10 responses) beginning at line 129. Record just the
results of each trial as “Y” or “N”
TRIAL 1
TRIAL 2
TRIAL 3
TRIAL 4
TRIAL 5
TRIAL 6
TRIAL 7
TRIAL 8
TRIAL 9
TRIAL 10
TRIAL 11
TRIAL 12
TRIAL 13
TRIAL 14
TRIAL 15
TRIAL 16
TRIAL 17
TRIAL 18
TRIAL 19
TRIAL 20
TRIAL 21
TRIAL 22
TRIAL 23
TRIAL 24
TRIAL 25
STEP 5:
ACTIVITY D:
Explain why each of the following simulations fails to model the real
situation properly:
1. Use a random integer from 0 through 9 to represent the number of heads that appear when
9 coins are tossed
2. A basketball player takes a foul shot. Look at a random digit, using an odd digit to
represent a good shot and an even number to represent a miss.
3. Simulate a baseball player’s performance at bat by letting 0 = out, 1 = single, 2 = double, 3 =
triple, 4 = home run.
4. The preferences for Dairy Queen Treats at a local franchise is as follows:
Peanut Buster Parfait = 38%
Banana Split = 42%
Dilly Bar = 14%
Other = 6%
Simulate the daily orders by assigning digits 0 – 3 to Peanut Buster Parfait, 4 – 7 to Banana
Split, 8 = Dilly Bar, and 9 = Other.
5. McDonald’s claims to have 60% of their business generated by drive-through orders.
Simulate a particular McDonald’s weekly orders by assigning even numbers to represent drive
through, and odd numbers to represent eat-in orders.
ACTIVITY E:
The owner of a bakery knows the daily demand for a highly perishable
cheesecake is as follows:
Number of cheesecakes sold
per day
0
Rel. freq.
0.05
1
0.15
2
0.25
3
0.25
4
0.20
5
0.10
1. Use simulation to find the demand for the cheesecake on 30 consecutive business days.
a) Assign digits to each cheesecake number:
no cheesecakes = ____________________
1 cheesecake = ___________________
2 cheesecakes = _____________________
3 cheesecakes = __________________
4 cheesecakes = _____________________
day:
5 cheesecakes = __________________
b) Perform the simulation for 30 days:
(Use line 135 on the random-digit table.)
day:
# cakes:
1
2
3
4
5
6
7
8
9
10
day:
# cakes:
11
12
13
14
15
16
17
18
19
20
day:
# cakes:
21
22
23
24
25
26
27
28
29
30
c) Summarize your findings:
# days of no cheesecakes =
# days of 1 cheesecake =
# days of 2 cheesecakes =
# days of 3 cheesecakes =
# days of 4 cheesecakes =
# days of 5 cheesecakes =
2. Suppose that it cost the baker $5 to produce a cheesecake, and that the unused
cheesecakes must be discarded at the end of the business day. Suppose also that the selling
price of a cheesecake is $13. Use the simulation from above to estimate the number of
cheesecakes that he should produce each day in order to maximize his profit.
If he bakes 0 cheesecakes each of the 30 days:
amount in sales: ________
cost for production: ________
what is his net profit? _________
If he bakes 1 cheesecake each of the 30 days:
amount in sales: ________
cost for production: ________
what is his net profit? _________
If he bakes 2 cheesecakes each of the 30 days:
amount in sales: ________
If he bakes 3 cheesecakes each of the 30 days:
amount in sales: ________
cost for production: ________
what is his net profit? _________
cost for production: ________
what is his net profit? _________
If he bakes 4 cheesecakes each of the 30 days:
amount in sales: ________
cost for production: ________
what is his net profit? _________
If he bakes 5 cheesecakes each of the 30 days:
amount in sales: ________
cost for production: ________
what is his net profit? _________
MAXIMUM PROFIT OCCURS WHEN _________________
Probability Models (6.2)
Learning Targets:
Explain how the behavior of a chance event differs in the short and long run
Explain what is meant by a “random phenomenon”
Explain the idea of probability being “empirical”
Define “probability” in terms of relative frequency
Define: sample space, event, equally likely outcomes, and independent events
Explain what is meant by a “probability model”
Construct a tree diagram
Use the multiplication principle to determine the number of outcomes in a sample space
Explain sampling with / without replacement
List the 4 rules that must be true for any assignment of probabilities
Explain what is meant by A  B and A  B
Explain what is meant by each of the regions in a Venn diagram
Give an example of 2 events, A and B, where A  B  
Use a Venn diagram to illustrate the intersection of two events A and B
Compute the probability of an event given the probabilities of the outcomes that make up the event
Compute the probability of an event in the special case of equally likely outcomes
Language of Probability:
A phenomenon is random if: a) _____________________________________________
and
b) ____________________________________________
The probability of any random phenomenon is ____________________________________
________________________________________________________________________
The idea of probability is empirical, or ___________________________________________
Probability Models:
Think about the event of tossing a coin. When we toss a coin, we cannot know the outcome in advance.
The description of a coin toss has two parts:
1) A list of all possible outcomes, which is called the ___________________________.
(each individual outcome is called an _______________)
2) A probability for each outcome.
If the above two parts are included in a mathematical description of a random phenomenon, then it is
called a ____________________________________.
EXAMPLE:
list the sample space, S, for the phenomenon and find the probability of each
outcome in the sample space:
a) a single coin toss
S = _____________
b) a roll of a die S = __________________
P = ___________
c) toss a coin and roll a die
P = ___________
S = _________________________________________________
P = _____________
Being able to properly list all outcomes in a sample space will be critical to determining
probabilities. There are three helpful techniques to make sure you don’t accidentally overlook
any outcomes:
TREE DIAGRAM
MULTIPLICATION PRINCIPLE
MAKE AN ORGANIZED LIST
TREE DIAGRAM: Make a tree diagram for tossing a coin and rolling a die
MULTIPLICATION PRINCIPLE: If you can do one task in n1 number of ways and a second
task in n2 number of ways, then both tasks can be done in n1 x n2 number of ways.
tossing a coin: n1 = _________
_______
rolling a die: n2 = __________
total outcomes =
MAKE AN ORGANIZED LIST:
Record the results of each of 4 tosses of a coin in order
EXAMPLE:
How many 3-digit numbers can you make?
(a) if no digits are repeated (called SAMPLING WITHOUT REPLACEMENT): ________________
(b) if digits may be repeated (called SAMPLING WITH REPLACEMENT): ___________________
Probability Rules:
1. Any probability is a number between 0 and Symbolically:___________________________
2. The sum of the probabilities of all possible outcomes must be 1 Symbolically: ____________
3. Probability of an event not occurring (called the _______________________ of the event) is 1
minus the probability that it does occur. Symbolically: _________________________
4. If 2 events have no outcomes in common (called ___________________ events or
________________________ events), then the probability that one or the other occurs is the sum
of their individual probabilities. Symbolically:______________________________
Probability Notation: We use set notation to describe events:
The event A  B is read _________________________________ and is the set of
___________________________________________________________. It is another way
to say ______________. The event
A B
is read ___________________________ and is
the set of ________________________________________________________. It is
another way to say ______________. The symbol

is used for __________________, which is
the event that _____________________________________.
Assigning Probabilities:
We can use a VENN DIAGRAM to help answer questions about probability. For example, suppose
that a certain high school has 90 students enrolled in AP Statistics, of which 16 are juniors. If their
junior class has 390 students, the high school has 1400 students enrolled, what is the probability that
a randomly selected student is a junior not taking AP Statistics?
In our Venn diagram, we should have overlapping circles (why?) and a border around the circles to
enclose space not in either circle:
AP Stats
Juniors
High School
We fill in the numbers given in the information and answer the question.
- FINITE NUMBER OF OUTCOMES –
If we have a finite number of outcomes we can use a table to help calculate probabilities.
EXAMPLE:
Faked numbers in tax returns, payment records, invoices, expense account
claims, and many other settings often display patterns that aren’t present in legitimate
records. Some patterns, like too many round numbers, are obvious and easily avoided by a
clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in
legitimate records often follow a distribution known as Benford’s law. Here it is:
First digit:
Probability:
1
0.301
Consider the events
2
0.176
3
0.125
4
0.097
5
0.079
6
0.067
7
0.058
8
0.051
9
0.046
A = {first digit is 1}
B = {first digit is 6 or greater}
C = {first digit is odd}
a) Find P(A) and P(B)
b) Find P  A  B 
c) Find the probability that a first digit is anything other than 1
d) Find P(C)
e) Find P(B or C)
- EQUALLY LIKELY OUTCOMES –
Assigning correct probabilities to individual outcomes often requires long observation of the
random phenomenon. But sometimes we are willing to assume equal likeliness because of a
balance in the phenomenon.
EXAMPLE: You roll a die. Find the probability that you roll a 2 or higher.
Event (roll):
Probability:
1
2
3
4
5
6
Independence and the Multiplication Rule:
We can find the probability that two events occur at the same time if the events are
independent.
Independent events are events in which the outcome of one event does not affect the
outcome of the other. Disjoint events cannot be independent. If A and B are disjoint, and if A
occurs, then B cannot occur.
In a toss of two coins, what is the probability that both coins land on heads?
Sample Space =
There are _________ possible outcomes. The combination H H is one of the possible outcomes. The
probability of H H occurring is __________. We can obtain this without writing out the sample
space. The Multiplication Rule tells us:
If A and B are independent events, then P(A and B) = P(A)P(B)
EXAMPLES:
1. You roll two dice. Find the probability that you roll doubles.
2. All human blood can be typed as one of O, A, B, or AB but the distribution of the types varies a bit
with race. Here is the distribution of the blood type of a randomly chosen black American:
Blood type:
Probability:
O
.59
A
.17
B
.18
AB
?
a) What is the probability of type AB blood? Why?
b) Maria has type B blood. She can safely receive blood transfusions from people with
blood types O and B. What is the probability that a randomly chosen black American
can donate blood to Maria?
3. A general can plan a campaign to fight one major battle or three small battles. He believes that
he has a probability of 0.6 of winning the large battle and probability 0.8 of winning each of the small
battles. Victories or defeats in the small battles are independent. The general must win either the
large battle or all 3 small battles to win the campaign. Which strategy should he choose?
General Probability Rules (6.3)
Learning Targets:
State the following rules: (A) the addition rule for disjoint events,
(B) the general addition rule for union of two events, (C) the general
multiplication rule for any two events.
Given any two events A and B, compute
P A  B
Define joint event, joint probability
Define independent events in terms of conditional probability
Given two events, compute their joint probability
Explain what is meant by the conditional probability
Use the general multiplication rule to define
P A | B
P B | A 
Review:
There are additional probability rules that govern any assignment of probabilities. We need
more rules so we can give probability models for more complex random phenomenon. Let’s
review the 5 rules we already have:
a. The value of any probability is ________________________________
b. The sum of the probabilities of the sample space is ________
c. If A and B are independent events, then P(A and B) = ____________
d. For any event A, P(AC) = ________________
e. If A and B are disjoint events, then P(A or B) = _______________
General Probability Rules:
Rule e applies to more than 2 disjoint events. In fact, it can extend to any number of disjoint
events.
The union of a collection of events is the event that any of them occur. If events A and B
are not disjoint, then ___________________________________________. The
probability of their union is _______ the sum of their probabilities.
Why are we using subtraction in this formula?
EXAMPLE:
Call a household “prosperous” if its income exceeds $100,000.
Call a household “educated” if the householder completed college. Select an
American household at random, and let A be the event that the selected
household is prosperous, and B be the event that the household is educated.
According to the Current Population Survey, P(A) = 0.138, P(B) = 0.261, and the
probability that a household is both educated and prosperous is P  A  B  = 0.082. What
is the probability that the household selected is either prosperous or educated.
Conditional Probability:
The probability we assign to an event can change if we know that some other event has
occurred. The other event that has occurred reduces the size of the sample space. So, when
calculating the probability, the denominator in the fraction has changed.
Consider rolling two dice and observing the sum. There are _________ possible outcomes for
the rolls, which is the denominator for any probability concerning the roll of two dice without a
condition. We wish to calculate the probability of rolling a sum of 8 at the same time one of
the dice shows a 3.
If we are given that one die already shows a 3, how many possible outcomes (of any sum) are
there now? _______ So the conditional probability of the sum being 8 given that one die is a
3 is:
The new notation P(A|B) or P(B|A) is a conditional probability. That is, it gives the
probability of one event (sum = 8) under the condition that we know another event (3 already
rolled).
EXAMPLE:
Students at the University of New Harmony received
10,000 course grades last semester. The grades in the table are
broken down by which school of the university taught the course.
The schools are Liberal Arts, Engineering and Physical Sciences, and
Health and Human Services.
Liberal Arts
Engineering & Physical Sciences
Health & Human Services
Total
A
2,142
368
882
3,392
B
1,890
432
630
2,952
Below B
2,268
800
588
3,656
Total
6,300
1,600
2,100
10,000
It is common knowledge that college grades are lower in engineering and the physical sciences
(EPS) than in the liberal arts and social sciences. Consider the two events:
A = the grade comes from an EPS course
B = the grade is below a B
Are these two events mutually exclusive? ____________
If we choose a grade at random, what is the probability that the grade will be below a B?
P(B) =
Find the probability that the randomly chosen grade is below a B given that the grade comes
from the EPS School.
Find the probability that a grade is an A given that it comes from a Liberal Arts course.
Extended Multiplication Rules:
The conditional probability rule can be re-written as a multiplication rule:
Notice how this is just a restatement of our original conditional probability rule.
This rule can be extended to more than two events and is used to calculate the probability of
the intersection. The intersection of a collection of events is the event that all of the events
occur at the same time.
Probability problems often require us to combine several of the basic rules into a more
elaborate calculation. If we use our organizational tools, the problems become easier to solve.
EXAMPLE:
Online chat rooms are dominated by the young. Teens are the biggest users.
If we look only at adult Internet users (aged 18 and over), 47% of the 18 – 29 age group chat
online, as do 21% of those aged 30 – 49, while just 7% of those 50 and over. To learn what
percent of all Internet users participate in chat rooms, we also need the age breakdown of
users. Ages 18 – 29 make up 29% of adult Internet users, 47% are age 30 – 49, and the
remaining 24% are over 50 years old.
a) What is the probability that a randomly chosen user of the Internet
participates in chat rooms?
b) What percent of adult chat room participants are aged 18 – 29?
Tree diagrams make the probability problem much simpler. In fact it takes longer to explain it
than it does to make the diagram and calculate the answer. Tree diagrams combine the
addition and multiplication rules.
Independent Events:
The conditional probability P(B|A) is generally not equal to the unconditional probability P(B).
This is because the occurrence of event A gives us some additional information about whether
or not event B occurs.
If knowing that event A occurs gives us no addition help toward whether or not B occurs, then
A and B are ______________________________________.
EXAMPLE:
Going back to the educated and prosperous households, where event A is
“prosperous” P(A) = 0.138, and event B is “educated” P(B) = 0.261, and the probability that a
household is both educated and prosperous is P  A  B  = 0.082. What is the conditional
probability that a household is prosperous given that it is educated?
Are the events prosperous and educated independent?
Chapter 6: Probability and Simulation: The Study of Randomness
Key Vocabulary:
Simulation
Trial
Random
Probability
Independence
random phenomenon
sample space
tree diagram
event
P(A)
replacement
AC
P(Ac)
disjoint
Venn diagram
union
intersection
joint event
joint probability
conditional probability
6.1 Simulation:
1. In statistics, what is meant by the term random?
2. In statistics, what is meant by probability?
3. In statistics, what is meant by an independent trial?
4. What is simulation?
5. Why do statisticians use simulation?
6. List the 5 steps for conducting a simulation.
6.2 Probability Models
1. What is probability theory?
2. In statistics, what is sample space?
3. In statistics, what is an event?
4. What is a probability model?
5. Explain why the probability of any event is a number between 0 and 1.
6. What is the sum of the probabilities of all possible outcomes?
7. Give your own example of an event that does not occur. What is it’s probability?
Explain.
8. What is meant by the complement of an event?
9. When are two events considered disjoint?
10. What is the probability of disjoint events both occurring?
11. Explain why the probability of getting heads when flipping a coin is 50%.
12. What is the multiplication rule for independent events?
13. Can disjoint events be independent? Explain.
14. If two events, A and B are independent, what must be true about AC and BC?
6.3 General Probability Rules
1. What is meant by the union of two or more events?
2. State the addition rule for disjoint events.
3. State the general addition rule for unions of two events.
4. Explain the difference between the rules in #2 and #3.
5. What is meant by joint probability?
6. What is meant by conditional probability?
7. State the general multiplication rule.
8. How is the general multiplication rule different from the multiplication rule for
independent events?
9. State the formula for finding conditional probability.
10. What is meant by the intersection of two or more events? Give an example by
drawing a Venn diagram.
11. Explain the difference between the union and the intersection of two or more
events.
12. Give an example of two disjoint events, and explain why they are disjoint.
Discrete and Continuous Random Variables (7.1)
Learning Targets:
Define: random variable, discrete random variable, continuous random variable,
probability distribution, density curve
Explain what is meant by a probability distribution and a uniform distribution
Construct the probability distribution for a discrete random variable
Construct a probability histogram for a discrete random variable
Introduction: Find the following probabilities on the roll of one die
P(1) _______
P(2) _______
P(3) _______
P(4) ________
P(5) _______
P(6) _______
Since we have several probabilities involved for the same event (the roll of a die), we could use
the notation P(X) where X represents the value that each roll could be. X is called a random
variable.
Definition:
Random Variable -
This chapter moves us from general probability to statistical inference.
Definition:
Statistical Inference –
Now our sample space, S, becomes a list of the possible values of the random variable instead
of a list of possible outcomes of an event. This section shows us 2 ways of assigning
probabilities to the values of the random variable.
Definition:
(a)
(b)
(c)
(d)
Discrete Random Variable -
EXAMPLE:
The instructor of a large class gives 15% each of A’s and D’s, 30% each of B’s
and C’s, and 10% F’s. Assign probabilities by making a probability distribution table. Use a
grade-point scale (A = 4, B = 3, etc.).
Find the probability that a student chosen at random
(a) receives a B or better
(b) doesn’t receive an F
(c) gets better than a D
Definition:
Continuous Random Variable -
(a)
(b)
(c)
(d)
(e)
EXAMPLE:
Use the density curve to find the following probabilities:
a) P(0.2 < X < 0.4) = ______________
1.0
0.75
b) P(0.2 < X < 0.4) = ______________
0.5
c) P(X = 0.2) = ______________
d) P(0 < X < 0.8) = ______________
0.25
0.2
0.4
0.6
0.8
1.0
1.2
1.4
We are most familiar with a density curve being the normal curve. Normal distributions are
probability distributions.
Remember N   ,   ? This means
________________________________________________
z
x 
is the formula for _____________________________________

Now, we use a capital letter X: z 
X 
represents a standard normal variable with N (0, 1)

We will use z-scores to calculate probabilities using table A. Your calculator has table A
programmed into it. It’s under the Distributions key (2nd VARS) “normalcdf”. You enter the
minimum and maximum z-scores for the area under the Normal curve.
EXAMPLE:
(a) find the following probabilities and (b) draw and shade the area under the
normal curve. Write down the calculator keystrokes used
Find P(z > 2.23) _______________________________________
Find P(z < -1.27) _______________________________________
Find P(-0.7 < z < 2.11) ____________________________________
EXAMPLE:
An opinion poll asks an SRS of 1500 American adults what they considered to be
the most serious problem facing our schools. Suppose that if we could ask all adults this
question, 30% would say “drugs”. This is the proportion for the entire population, p. Our
sample proportion is an estimate of p, so we call it
particular
p
p.
We will see in a later chapter that this
has a distribution of N(0.3, 0.0118). What is the probability that the poll result
differs from the truth about the population by more than 2 percentage points?
Means & Variances of Random Variables (7.2)
Learning Targets:
Define “mean of a random variable”
Calculate the mean, variance, and standard deviation of a discrete random
variable, 
, 
, 2
,  2X  Y
a  bX
XY
a  bX
Explain: the Law of Large Numbers, the Law of Small Numbers, standard
deviations of combined random variables,
ACTIVITY:To investigate means of a random variable, consider a random variable that
takes values {0, 1, 2, 3, 5, 8}. Complete the following:
1. Calculate the mean of the population, 
2. Make a list of all of the samples of size 2 from this population. (There should be 15)
3. Find the mean of each of the 15 samples of size 2.
4. Find the mean of the 15 x values. Compare this mean to the mean you calculated in
step 1
Population mean
Sample#
 = _____________
Sample
0, 1
x
0.5
2
0, 2
1
3
0, 3
1
Sample#
6
Sample
x
1, 2
Sample#
11
7
1, 3
12
8
1, 5
13
4
9
14
5
10
15
Sample
mean of all x s = ________
x
represents ___________________________________________________
 x represents _____________________________________________________
Since the value of a random variable is a numerical outcome of a random phenomenon, the
probabilities can all be different. To find the mean value, we apply a formula.
Formula:
X is a discrete random variable with the distribution:
X
P(X)
x =
x1
p1
x2
p2
x3
p3
…
…
xk
pk
x
EXAMPLE:
According to the following Census Bureau data, what is the mean size of an
American household?
Inhabitants
Proportion of households
1
.25
2
.32
3
.17
4
.15
5
.07
6
.03
7
.01
x =
For continuous random variable distributions,
 x is the point at which the area under the
curve would ________________________ if it were made out of a solid material. If the
density curve is symmetric, the mean is the center. If the density curve is skewed, we need
advanced mathematics to find the mean.
Law of Large Numbers:
As the number of observations increases (sample size), the
mean x eventually approaches the mean  of the population.
How many observations are necessary? That depends on the ______________________ of
the random outcomes. The more variable the outcomes, the more trials are needed.
Rules for Means:
1.
2.
EXAMPLE:
Gain Communications sells aircraft communications units to both the military
and civilian markets. Next year’s sales depend on market conditions that cannot be predicted
exactly. Gain follows the modern practice of using probability estimates of sales. Gain makes
a profit of $2000 on each military unit sold and $3500 on each civilian unit sold.
Let X be the random variable: # units sold in military division.
Let Y be the random variable: # units sold in civilian division.
Then: X
P(X)
1000
.1
3000 5000 10,000
.3
.4
.2
Y
P(Y)
300
.4
500
.5
750
.1
(a) Find the mean number of military units sold and civilian units sold
MILITARY:
CIVILIAN:
(b) Find the mean number of units sold overall:
(c) Find the mean profit from military sales and the mean profit from civilian sales; then find
the overall mean profit. (Military profit = $2000 each; Civilian profit = $3500 each)
MILITARY:
CIVILIAN:
OVERALL:
Mean is a measure of ______________. Variance and standard deviation are measures of
________________________.
Formula:
X is a discrete random variable with the distribution:
X
P(X)
x2 =
x =
x1
p1
x2
p2
x3
p3
…
…
xk
pk
with mean =
x
EXAMPLE:
Find the standard deviation for Gains Communications sales in the military
division.
X
P(X)
1000
.1
3000
.3
5000
.4
10,000
.2
Y
P(Y)
300
.4
500
.5
750
.1
Let the calculator do the calculations for you.
Rules for Variances:
1.
2.
EXAMPLE:
The payoff, X, of a $1 ticket in the Tri-State Pick 3 game is $500 with
probability 0.001 and $0 the rest of the time.
(a) Find the distribution of X
(b) Find X and X
(c) Find your average winnings
(d) Suppose you buy a $1 ticket on each of 2 different days. Calculate your mean total payoff
and the standard deviation of the total payoff.
Chapter 7: Random Variables
Key Vocabulary:
Random variable
Law of Large Numbers
Discrete random variable
Probability distribution
standard deviation
Probability histogram
Density curve
continuous random variable
variance
uniform distribution
normal distribution
Probability density curve
X
Y
expected value
7.1 Discrete and Continuous Random Variables:
1. What is a discrete random variable?
2. If X is a discrete random variable, what information does the probability
distribution of X give?
3. In a probability histogram, what does the height of each bar represent?
4. In a probability histogram, what is the sum of the heights of each bar?
5. What is a continuous random variable?
6. If X is a continuous random variable, how is the probability distribution of X
described?
7. If X is a discrete random variable, do P(X > 2) and P(X > 2) have the same value?
Explain.
8. If X is a continuous random variable, do P(X>2) and P(X>2) have the same value?
Explain.
9. How is a normal distribution related to a probability distribution?
10. Is a probability distribution always a normal distribution? Explain.
7.2 Means and Variances of Random Variables:
1. Explain the difference between the notations x and X .
2. What is meant by the expected value of X?
3. How do you calculate the mean of a discrete random variable X?
4. Explain the Law of Large Numbers.
5. Given the mean X and  Y explain how to calculate the mean X  Y .
6. Given the mean X and  Y explain how to calculate the mean X  4Y .
7. Explain how to calculate the variance of a discrete random variable.
8. Given the variance of a discrete random variable, explain how to calculate the
standard deviation.
9. Suppose X and Y are independent random variables, given  2 X and  2 Y explain how
to calculate  2 X  Y and  X  Y .
10. Suppose X and Y are independent random variables, given  2 X explain how to
calculate  23  2X and 3  2X .
Binomial Distributions (8.1)
Learning Targets:
Describe the conditions that need to be present to have a binomial
setting
Define binomial distribution
Explain what is meant by the sampling distribution of a count
Explain the difference between binompdf (n, p, x) and binomcdf(n, p,
x)
State the mathematical expression that gives the value of a binomial
coefficient.
State the mathematical expression used to calculate the value of
binomial probability
Evaluate a binomial probability by using the mathematical formula for
P(X=k).
Use your calculator to help evaluate a binomial probability
Calculate the mean and variance of a binomial distribution
Review:
Discrete Probability Distribution - ___________________________________________
Continuous Probability Distribution - __________________________________________
Binomial Random Variable:
A random variable can also be what is called a binomial random variable if the data are
produced in a binomial setting. There are 4 conditions that determine a binomial setting:
a)
b)
c)
d)
X = number of successes, and is called a binomial random variable because the number of
successes varies according to the random phenomenon. We use the notation __________ to
tell you the distribution is binomial with “n” observations each having probability “p”.
EXAMPLES:
Determine whether the following situations are binomial or not:
1) Deal 10 cards from a shuffled deck and count the number of red cards dealt.
CHECK THE REQUIREMENTS:
a)
c)
b)
d)
If binomial, X represents the number of successes, which is _________________________
2) A couple decides to continue having children until their first girl is born. X is the number of
children they have.
CHECK THE REQUIREMENTS:
a)
c)
b)
d)
If binomial, X represents the number of successes, which is __________________________
3) A quality engineer selects an SRS of 10 switches from a large shipment for detailed
inspection. Unknown to the engineer, 10% of the switches in the shipment fail to meet the
specifications. X is the number of detective switches in the sample.
CHECK THE REQUIREMENTS:
a)
c)
b)
d)
If binomial, X represents the number of successes, which is __________________________
Binomial Probabilities:
Once we determine the type of distribution we have, we will be asked to calculate probabilities.
Use example (3) to find the probability that out of the 10 switches inspected, exactly 2
switches in the sample will fail the inspection.
Determine what event is a “success” ________________________________
Determine what the random variable X represents:
X = _______________________________________
P(X) = ______________________________________
Make a probability distribution:
X
P(X)
P(X = 0) is the probability that no switches fail.
P(X = 1) is the probability that exactly one switch fails. (We don’t know which switch it will be)
In order to calculate P(X = 2):






Number of ways
we can have 2
failing switches


Probability
of a
switch
failing
Probability of a switch not failing
n
nk
P(X  k)    p k 1  p 
k 
Probability Formula:
We can also use the Binomial pdf (probability distribution function) to calculate the
probabilities.
Binompdf (n, p, X):
Binompdf (total # trials, probability of success, X-value)
If we wanted to calculate the probability that no more than 2 switches failed, we would use the
cumulative distribution function (cdf)
Binomcdf (n, p, X):
Binomcdf (total # trials, probability of success, X-value)
EXAMPLE:
Corinne is a basketball player who makes 75% of her free throws over the
season. In a key game, Corinne shoots 12 free throws and makes only 7 of them. The fans
think she failed because she was nervous. Is it unusual for Corinne to perform this poorly?
What question are we really answering? _________________________________________
Identify:
Success: _______________________________________________
X represents ___________________________________________
n = ________
p = _________
X = _________
pdf or cdf? ______________
Calculate:
Binomial Mean and Standard Deviation:
What would we guess to be the average (mean) number of shots made by Corinne in games
where she shoots 12 free throws?
Mean of a Binomial Random Variable:
Standard Deviation of a Binomial Random Variable:
 n p
  n p 1  p
EXAMPLE:
Among employed women, 25% have never been married. Select 10 employed
women at random.
(a) What is the distribution of X?
(b) What is the probability that exactly 2 of the 10 women in your sample have never been
married?
(c) What is the probability that 2 or fewer have never been married?
(d) What is the probability that 6 or more have never been married?
(e) What is the probability that more than 4 have never been married?
(f) What is the mean number of women in such a sample who have never been married?
(g) What is the standard deviation?
Geometric Distributions (8.2)
Learning Targets:
Review:
Describe what is meant by a geometric setting
Given the probability of success, p, calculate the probability of
getting the first success on the nth trial.
Calculate the mean (expected value) and the variance of a geometric
random variable
Calculate the probability that it takes more than n trials to see the
first success for a geometric random variable
Conditions for a Binomial Distribution:
a)
b)
c)
d)
Geometric Random Variable:
A random variable can also be what is called a geometric random variable if the data are
produced in a geometric setting. There are 4 conditions that determine a geometric setting:
a)
b)
c)
d)
X = number of trials to obtain the first success.
EXAMPLES:
Determine if the following situations are geometric or not:
1) Deal 10 cards from a shuffled deck and count the number X of red cards dealt.
CHECK THE REQUIREMENTS:
a)
c)
b)
d)
If geometric, X represents the trial where the first success occurs.
2) A couple decides to continue having children until their first girl is born.
CHECK THE REQUIREMENTS:
a)
c)
b)
d)
If geometric, X represents the trial where the first success occurs. ________________
3) Blood type is inherited, but children inherit independently of each other. Count the
number of type A blood among 5 children
CHECK THE REQUIREMENTS:
a)
c)
b)
d)
If geometric, X represents the trial where the first success occurs. ________________
Geometric Probabilities:
A game consists of rolling a single die. The event of interest is rolling a 3. We can label each
roll as one of two possible outcomes: success = ___________________________;
failure = _____________________________. To calculate the probabilities,
we consider each roll separately:
X = 1: P(X = 1) finds the probability of a success on the first roll, which is _______________
X = 2: P(X = 2) finds the probability of a success on the second roll, which means:
roll #1 was a failure (Probability = ______________)
roll #2 was a success (Probability = _____________)
so P(X = 2) = _________________
X = 3: P(X = 3) finds the probability of ________________________________________
roll # 1 was a ______________________ (Probability = ______________)
roll # 2 was a ______________________ (Probability = ______________)
roll # 3 was a ______________________ (Probability = ______________)
so P(X = 3) = ____________
Probability Formula:
P(X  n)  1  p 
n 1
p
We can also use the Geometric pdf (probability distribution function) to calculate the
probabilities.
Geometpdf (n, p)
Geometpdf (probability of success, trial # of 1st success)
EXAMPLE:
Find the probability that the first 3 occurs on the 5th roll.
If we wanted to find the probability that the first roll of a 3 occurred sometime before the
nth roll, we would use the cumulative distribution function
Geometcdf (n, p)
Geometcdf (probability of success, maximum trial # of 1st success)
EXAMPLE:
Bob is a basketball player who makes 65% of his free throws over the season.
We put him on the free-throw line and ask him to shoot free throws until he misses one.
Let X = # free throws Bob makes until he misses. Construct a probability distribution table.
X
P(X)
1
2
3
4
5
(a) Find the probability that Bob’s first miss occurs on his 5th shot
(b) Find the probability that Bob’s first miss occurs before his 5th shot
(c) Find the probability that Bob’s first miss occurs after the 5th shot
Geometric Mean and Standard Deviation:
Mean of a Geometric Random Variable:
1
p
Variance of a Geometric Random Variable:
 2  1 2p
p
Standard Deviation of a Geometric Random Variable:
6
etc
EXAMPLE:
Suppose that Albert, a well-known major league baseball player, finished last
season with a 0.325 batting average. He wants to calculate the probability that he will get his
first hit of this new season at his first at-bat. He also wants to know his expected number of
at-bats until he gets a hit.
EXAMPLE:
In 1986-1987, Cheerios cereal boxes displayed a dollar bill on the front of the
box and a cartoon character who said, “Free $1 bill in every 20th box.” Conduct a simulation
to determine the number of boxes of Cheerios you would expect to buy in order to get one of
the “free” dollar bills.
Success = _____________________ Digits: ____________________
Failure = ______________________ Digits: ____________________
Use the following random digits. The trial is over as soon as you find a $1 bill.
43909 99477 25330 64359 40085 16925 85117 36071
15689 14227 06565 14374 13352 49367 81982 87209
36759 58984 68288 22913 18638 54303 00795 08727
According to Cheerios, how many boxes would we expect to have to open? _______________
How many did our simulation say we have to open? ________________
Calculate the variance and the standard deviation using the probability from Cheerios. How
does the standard deviation help explain the results of our simulation?
Chapter 8: The Binomial and Geometric Distributions
Key Vocabulary:
Binomial setting
Binomial distribution
Probability distribution function (pdf)
Binomial coefficient
Binomial random variable
B (n, p)
Cumulative distribution function (cdf)
n!
n
   k ! n 1 !
 
k 
8.1 The Binomial Distributions:
1. What are the four conditions for the binomial setting?
2. In the binomial distribution, what do the parameters n and p represent?
3. What is meant by B(n, p)?
4. What is the difference between a probability distribution function (pdf) and a
cumulative distribution function (cdf)?
5. How do we find the mean and standard deviation for a binomial random variable?
6. There are 50 poker chips in a container, 25 of which are red, 15 white, and 10 blue.
You draw a chip without looking 25 times, each time returning the chip to the container.
a) What is the probability that you will draw 9 or fewer blue chips?
b) What is the probability that you will draw 6 or more red chips.
8.2 The Geometric Distributions:
1. What are the four conditions for the geometric setting?
2. Explain the difference between the binomial setting and the geometric setting.
3. How do you calculate the mean of a geometric random variable X?
4. How do you calculate the variance of a geometric random variable?
5. Explain what has been the most difficult part of this class so far.
Sampling Distributions (9.1)
Learning Targets:
Compare and contrast parameter and statistic
Explain what is meant by sampling variability
Define the sampling distribution of a statistic
Explain how to describe a sampling distribution
Define an unbiased statistic and an unbiased estimator
Describe what is meant by the variability of a statistic.
Explain how bias and variability are related to estimating with a sample
Introduction:
The remainder of this course focuses on Statistical Inference. We will
be asking “how often will this method give a correct answer if I used it many, many
times?” This chapter prepares us for the study of statistical inference by looking at the
probability distributions of sample proportions and sample means.
Parameters and Statistics:
A recent poll asked random individuals “Are you afraid to
go outside at night?” The results showed that 45% of the sample said yes, they were afraid to
go outside at night. Does that mean 45% of all people are afraid to go outside at night?
45% describes the sample; it is used to estimate the _______________________________.
EXAMPLE A:
State whether each boldface number is a parameter or a statistic:
1. The Tennessee STAR experiment randomly assigned children to regular or small classes during their
first four years of school. When these children reached high school, 40.2% of blacks from small classes
took the ACT or SAT college entrance exams. Only 31.7% of blacks from regular classes took one of
these exams.
2. A random sample of female college students has a mean height of 64.5 inches, which is greater than
the 63-inch mean height of all adult American women.
Sampling Distributions:
Suppose I wanted to know what percent of the class has a dog
for a pet.
Population: _____________________________________________________________
Population Proportion: ________________________
Now, I choose an SRS of 10 students and survey them. I can calculate the statistic:
p1 
# yes
 %
10
Then I choose a different SRS of 10 students and calculate a new statistic:
p2 
# yes
%
10
And I choose yet another SRS of 10 students and calculate a 3rd statistic, p 3 .
Are these 3
statistics always equal?
p 4 = ___________
p 5 = ______________
p 6 = ______________
Every possible combination of 10 combine to create the sampling distribution.
Since the value of the statistic varies with repeated sampling, a graph of all combinations would
be the graph of the sampling distribution.
EXAMPLE B:
Let us illustrate the idea of sampling variability and a sampling distribution in
the case of a very small sample from a very small population. The population is the scores of 10
students on an exam:
Student #
Score
0
82
1
62
2
80
3
58
4
72
5
73
6
65
7
66
8
74
9
62
The parameter of interest is the mean score in this population, which is 69.4. The sample is an
SRS of size n = 4 drawn from the population. Because the students are labeled 0 to 9, a single
random digit from Table B chooses one student for the sample.
a) Use table B (line 120) to draw an SRS of size 4 from this population. Write the four scores
in your sample and calculate the mean x of the sample scores. This statistic is an estimate of
the population parameter.
LINE 120 (STUDENT #):
STUDENT TEST SCORE:
3
5
4
7
x 1 = _____________
b) Continue in line 120, wrapping around to line 121, and repeat this process 9 more times.
SCORES:
SCORES:
x 2 = _____________
x 3 = _____________
SCORES:
SCORES:
x 4 = _____________
x 5 = _____________
SCORES:
SCORES:
x 6 = _____________
x 7 = _____________
SCORES:
SCORES:
x 8 = _____________
x 9 = _____________
SCORES:
x 10 = _____________
Notice the sampling variability in all the x values. Find the mean of all 10 x values _________
The sampling distribution is the ideal pattern that would emerge if we looked at all possible
samples of size n from our population.
The Bias of a Statistic:
How trustworthy is a statistic as an estimate of a parameter?
We use the concept of bias and describe the bias of a statistic as the center of the sampling
distribution rather than the bias of a sampling method.
The Variability of a Statistic:
The sample proportion from a random sample of any
size is an unbiased estimator of the population parameter, but larger samples have a clear
advantage. Since there is less variability among large samples than among small samples, the
large samples are more likely to produce an estimate close to the true value of the parameter.
The variability of a statistic does not depend on the size of the population.
Bias and Variability:
We can think of the true value of the population parameter as the
bull’s eye on a target and think of the sample statistic as an arrow fired at the target. Both
bias and variability describe what happens when we take many shots at the target.
Bias means that __________________________________________________________
_______________________________________________________________________
High variability means that __________________________________________________
_______________________________________________________________________
Sample Proportions (9.2)
Learning Targets:
Describe the sampling distribution of a sample proportion
Compute the mean and standard deviation for the sampling distribution
Identify the “rule of thumb” that justifies the use of the formula for the
standard deviation
Identify the conditions necessary to use a Normal approximation for the
sampling distribution
Use a Normal approximation for the sampling distribution of
p to solve
probability problems
The Sampling Distribution of a Sample Proportion p :
To describe the sampling proportion of p , we need to discuss the center, shape, and spread.
The center uses the mean value. In a sampling distribution of sample proportions (all possible
samples of the same size), the mean of the distribution of p is the parameter p.
When discussing the spread, if the population is much larger than the sample, the standard
deviation of the distribution of p is  
p 1  p 
n
. This formula for standard deviation only
applies if the population is __________________________________________________.
YOU MUST CHECK THIS CONDITION (AND SHOW THE CHECK) BEFORE USING THIS
FORMULA.
Using the Normal Approximation for p :
To discuss the shape, remember that
we’re talking about the distribution of all possible samples. We said that the larger the sample
size, the less variability we will have. If the sample size is large enough, we will have a shape
close to Normal. How large is “large enough”?
if:
np > 10
and if:
n(1-p) > 10
we use the Normal approximation
YOU MUST ALWAYS CHECK THIS CONDITION (AND SHOW THE CHECK) BEFORE
ASSUMING YOU HAVE A NORMAL DISTRIBUTION
EXAMPLE A:
You ask an SRS of 1500 first-year college students whether they applied for
admission to any other college. Actually, 35% of all first-year students applied to colleges
besides the one they’re attending. What is the probability that your sample will give a result
within 2 percentage points of the true value?
n = ______ p = ______
* Can we assume a Normal distribution?
* Can we use our formula for standard deviation?
EXAMPLE B:
The Gallop Poll once asked a random sample of 1540 adults, “Do you happen to
jog?” Suppose that in fact 15% of all adults jog.
a) Find the mean and, if you can, the standard deviation of the sample proportion p of the
proportion who jog.
b) Check that you can assume a Normal approximation for the distribution of p .
c) If you can use a Normal distribution, find the probability that between 13% and 17% of the
sample jog.
d) What sample size would be required to reduce the standard deviation of the sample
proportion to one-half the value you found in (a)?
Sample Means (9.3)
Learning Targets: Given the mean and standard deviation of a population, calculate the mean and
standard deviation for the sampling distribution of a sample mean.
State the Central Limit Theorem
Identify the shape of the sampling distribution of a sample mean drawn from
a population that has a Normal distribution
Use the Central Limit Theorem to solve probability problems for the sampling
distribution of a sample mean
Introduction:
We use sample proportions most often when we are interested in categorical variables. We
look at “what proportion of ...” or “what percent of adults ...”
When we record quantitative variables we are interested in other statistics, like mean and
standard deviation. Sample means are among the most common statistics. A sample
distribution for the sample mean would describe the means of
__________________________________. Since the value of the sample mean depends on
the sample we use, the sample mean is a random variable.
The Mean and the Standard Deviation of x :
We will be using statistics calculated from samples to make decisions about the population. To
estimate the population mean, we can use the sample mean because:
The mean of the sample means is the population mean.
x  
If the sample size is increased, the variability of the sample mean decreases.
x  
n
(only use this formula when the population is at least 10 times as large as the sample.)
The distribution of the sample mean becomes closer to a normal distribution as the sample size
becomes larger, regardless of the shape of the population from which the sample is drawn.
(The Central Limit Theorem)
EXAMPLE:
Suppose a population has a mean of 30 and a variance of 25. If a sample size of
100 is drawn from the population, what is the probability that the sample mean will be larger
than 31?
EXAMPLE:
A company that owns and services a fleet of cars for its sales force has found
that the service lifetime of the disc brake pads varies from car to car according to a Normal
distribution with mean   55,000miles and standard deviation   4500miles . The company
installs a new brand of brake pads in 8 cars.
a) If the new brand has the same lifetime distribution as the previous type, what is the
distribution of the sample mean lifetime for the 8 cars? (center, shape, spread)
b) The average life of the pads on these 8 cars turns out to be x  51,800 miles. What is the
probability that a sample mean lifetime would be 51,800 or less if the true lifetime distribution
is unchanged? (This probability is evidence of whether or not the new brand of pads has a lifetime less
than 55,000 miles)
Chapter 9: Sampling Distributions
Key Vocabulary:
Parameter
Statistic
Sampling Variability
Law of Large Numbers
Sampling Distribution
Unbiased
Central Limit Theorem
Population mean
Sample mean
x vs.
 ; p vs. p
9.1 Sampling Distributions
1. Explain the difference between a parameter and a statistic.
2. Explain the difference between p and p .
3. Explain the difference between p and x .
4. What is meant by the sampling distribution of a statistic?
5. When is a statistic considered unbiased?
6. How is the size of the sample related to the spread of the sampling distribution?
9.2 Sample Proportions
1. In an SRS of size n, what is true about the sampling distribution of p when the
sample size, n, increases?
2. In an SRS of size n, what is the mean of the sampling distribution of p ?
3. In an SRS of size n, what is the standard deviation of the sampling distribution of
p?
4, What happens to the standard deviation of p as the sample size, n, increases?
5. When does the formula  
p 1  p 
n
apply to the standard deviation of p ?
6. When the sample size, n, is large, the sampling distribution of p is approximately
normal. What test can you use to determine if the sample is large enough to assume
that the sampling distribution is approximately normal?
7. Explain the difference between “rule of thumb 1” and “ rule of thumb 2”.
9.3 Sample Means
1. What symbols are used to represent the parameters in this section?
2. What symbols are used to represent the statistics in this section?
3. Since averages are less variable than individual outcomes, what is true about the
standard deviation of the sampling distribution of x ?
4. If we draw an SRS of size n from a population that has a Normal distribution with
mean  and standard deviation  . Give three characteristics of the sample mean, x .
5. What does the Central Limit Theorem say about the shape of the sampling
distribution of x ?
6. What is your plan in preparing for the final exam?
Estimating with Confidence (10.1)
Learning Targets:
List the 6 basic steps in the reasoning of statistical estimation
Distinguish between a point estimate and an interval estimate
Identify the basic form of all confidence intervals
Explain what is meant by margin of error
State in nontechnical language what is meant by a level C confidence interval
State the 3 conditions that need to be present in order to construct a valid confidence interval
Explain what it means by the “upper p critical value” of the standard Normal distribution
For a known population standard deviation, construct a level C confidence interval for a population mean
List the 4 necessary steps in the creation of a confidence interval
Identify 3 ways to make the margin of error smaller when constructing a confidence interval
Once a confidence interval has been constructed for a population value, interpret the interval in the context of the
problem
Determine the sample size necessary to construct a level C confidence interval for a population mean with a
specified margin of
error
Identify as many of the 6 warnings about constructing confidence intervals as you can
Introduction:
The goal of statistical inference is to use the sample data to _________________________
__________________. We won’t be sure that our conclusions are correct – a different
sample might lead to _____________________________. Using probability to express the
strength of our conclusions will add support to our decisions. In this chapter, we will
___________________ the value of a population parameter with a confidence interval.
Confidence Intervals:
Let’s say we have an SRS of size 840 with a mean test score of 272 points. What would we
guess the population mean,  , to be? _________ If we had every possible sample of size 840
(the “sampling distribution”), what would we know?
*_____________________________________________________________________
*_____________________________________________________________________
* _____________________________________________________________________
Realistically we wouldn’t know  if we don’t know  . But let’s suppose  = 60 for this
population. Then where are 95% of all samples?
When calculating a confidence interval, there are only 2 possible results:
* _____________________________________________________________________
* _____________________________________________________________________
The probability that our confidence interval contains the true mean,  , is either _______ or
_______.
All confidence intervals are in the form:
ESTIMATE

MARGIN OF ERROR
(Obtained from the sample) (how far from the sample we are willing to go)
The 68-95-99.7 rule will work if we want to be
68% confident: Margin of error = _________________________
If we repeat this process over and over, 68% of the intervals will contain the true mean.
95% confident: Margin of error = _________________________
If we repeat this process over and over, 95% of the intervals will contain the true mean.
99.7% confident: Margin of error = _________________________
If we repeat this process over and over, 99.7% of the intervals will contain the true
mean.
What do we need to find if we want a 90% confidence interval?
When 90% of the area is in the middle of the curve that means 10% is left out. So each tail
has 5% of the area. We need to find the point where that happens.
Each * is a certain number of standard deviations away from  . We call it z* and the value of
z* depends on the confidence level we wish to use.
Confidence level
90%
95%
99%
EXAMPLE 1:
tail area
0.05
0.025
0.005
z* value
1.645
1.960
2.576
Find z* for 80% confidence level
___________________
Find z* for 98% confidence level
___________________
A level “C” confidence interval for  is:
  

 n
xz*
EXAMPLE 2:
A study of the career paths of hotel general managers sent questionnaires to an SRS of 160
hotels belonging to major US hotel chains. There were 114 responses. The average time those
general managers had spent with their current company was 11.78 years. Give a 99% confidence
interval for the mean number of years general managers of major chain hotels have spent with
their current company. (use  = 3.2 years)
For all confidence intervals:
* The user chooses the confidence level when choosing the sampling method and setting up the
experiment; the margin of error depends on the confidence level.
* Margin of Error =
  

 n
z*
Numerator = z* and 
Denominator =
n
* We would like a high confidence level and a small margin of error
* Margin of error gets smaller when:
(how does a fraction get smaller?)
* _____________________________________________________
* _____________________________________________________
* _____________________________________________________
EXAMPLE 3: How large of a sample of the hotel managers in the previous example
would we need to estimate the mean  within  1 year with 99% confidence?
Estimating a Population Mean (10.2 A)
Learning Targets:
Identify the 3 conditions that must be present before estimating a population mean
Explain what is meant by the standard error of a statistic in general and by the standard error of the
sample mean in particular
List 3 important facts about the t-distributions. Include comparisons to the standard Normal curve.
Use Table C to determine critical t-values for a given level C confidence interval for a mean and a
specified number of degrees of freedom
Construct a one-sample t confidence interval for a population mean (remembering to use the 4-step
procedure)
Conditions for Inference about a Population Mean:
As mentioned in section 10.1, we usually will not know the population standard deviation when
finding confidence intervals. If  is not known, we cannot calculate the standard deviation for
the sampling distribution,  x . We must estimate  from the data, even though we are
primarily interested in  . Before we can estimate, we need to verify 3 important
conditions:
*
*
*
The
t
Distribution:
Once we have verified the conditions, we can estimate  with “s”, the sample standard
deviation. Then the standard deviation of the sampling distribution,  x , becomes the standard
_____________________ of the sample mean. Since  was replaced by s, the statistic has
more ________________ and no longer has a Normal distribution. We cannot find the
standardized value, z. There is another standardized value we can use, called ______. The
formula is very similar to the z-score formula:
Like z, this standardized value tells us the number of standard deviation units x is from  .
Unlike z, there is a different t-distribution for each _______________________. We
specify a particular t-distribution by giving its ___________________________________.
The notation is ______________. Table C in the back of the book gives critical values t* for
the t-distributions. (t* tells us ____________________________________________)
Each row in the table contains critical values for one of the t-distributions; the degrees of
freedom appear at the left of the row. By looking down any column, you can check that the tcritical values approach the normal values as the degrees of freedom increase. The t-table
uses area to the right of t*.
EXAMPLE 1:
Find the critical value t* from table C to satisfy each of the following
conditions:
a) The t-distribution has 5 degrees of freedom and probability 0.05 to the right of t*. ____
b) The t-distribution has 21 degrees of freedom and probability 0.99 to the left of t*. ____
c) Used for a 95% confidence interval based on 10 observations (n = 10). __________
d) Used for a 99% confidence interval from an SRS of 20 observations. __________
e) Used for an 80% confidence interval from a sample size of 7. __________
Facts About the t-Distribution:
*
The density curves of the t-distributions are similar in shape to the _________________
______________________________. They are symmetric about _____________, singlepeaked, and ________________________.
*
The spread of the t-distributions is a bit _____________________ than that of the
standard normal distribution. The t-distributions have more area/probability in ___________
and less in the _________________ than the standard normal distribution does. This is true
because _______________________________________________________________.
*
As the degrees of freedom (k) increase, the t(k) density curve approaches the _________
_________________________more closely. This happens because _________________
___________________________________________________________________.
The One-Sample
t
Confidence Intervals:
Now we use this knowledge to construct confidence intervals for an unknown mean, with an
unknown standard deviation:

s 

 n
x  t*
Steps to displaying a confidence interval:
P:
“Let ________ represent the mean _________________ of all _______________.”
A:
N:
I:
C:
“We are ______% confident that the true mean __________________ is between
_____ & _____.”
(Being _______% confident means that since ______% of all intervals created by this method
will contain the true mean, we are pretty sure our interval is one of those intervals)
EXAMPLE 2:
Poisoning by the pesticide DDT causes tremors and convulsions. In a study of
DDT poisoning, researchers fed several rats a measured amount of DDT. They then made
measurements on the rats’ nervous systems that might explain how DDT poisoning causes
tremors. One important variable was the “absolutely refractory period”, the time required for
a nerve to recover a stimulus. This period varies normally. Measurements on four rats gave the
data below (in milliseconds).
1.6
1.7
1.8
1.9
a) Find the mean refractory period ( x ) and the standard error of the mean(Sx).
b) Give a 90% confidence interval for the mean absolutely refractory period for all rats of
this strain when subjected to the same treatment.
P
A
N
I
C
Robustness of
t
- procedures:
One-sample t-procedures are exactly correct when the population is ___________________.
However, no real data are exactly Normal. Procedures that are robust are not strongly
influenced by the lack of normality. If there are no outliers in the data, the t-procedures
can be used if the population isn’t Normal. The t-procedures are strongly influenced by
__________________. We assume that the population is
___________________________ in order to justify the use of t-procedures. If there are
_____________________ and the sample size is small, the results will not be reliable. For
small sample sizes, sizes less than _______, only use t-procedures if the data are close to
Normal. If the sample size is larger, at least _______, we can use t-procedures if there are
no outliers or ____________________. If the sample is size is large enough, at least
________, we can safely use t-procedures, even if the data is skewed.
Estimating a Population Mean (10.2 B)
Learning Targets:
Describe what is meant by “paired-t procedures”
Calculate a level C confidence interval for a set of paired data
Explain what is meant by a robust inference procedure and comment on the robustness of
t-procedures
Discuss how sample size affects the usefulness of t-procedures
Paired
t
Procedures:
Comparative studies are more convincing than single-sample investigations, so ____________
inference is not as common as comparative _______________________. In a comparative
study, we may want to compare two _____________, or we may want to compare two
___________________. In either case, the samples must be chosen __________________
and _____________________________ in order to perform statistical inference. Because
matched pairs are not chosen independently, we will not use two-sample inference for a
matched pairs design. Instead, we apply the one-sample _____________________ to the
observed ____________________. If the same subjects are creating both data sets, it
is a matched pairs design.
EXAMPLES
The following situations require inference about a mean or means. Identify each as a singlesample, a matched-pairs sample, or a two-sample:
A) An education researcher wants to learn whether it is more effective to put questions
before or after introducing a new concept in an elementary school mathematics text. He
prepares two textbook segments that teach the same concept, one with motivating questions
before and the other with review questions after. He uses each text segment to teach a
separate group of children. The researcher compares the scores of the groups on a test over
the material.
B) Another researcher approaches the same issue differently. She prepares textbook
segments on two unrelated topics. Each segment comes in two versions, one with questions
before and the other with questions after. The subjects are a single group of children. Each
child studies both topics, one (chosen at random) with questions before and the other with
questions after. The researcher compares test scores for each child on the two topics to see
which topic he or she learned better.
The parameter in a matched-pair t-procedure is either:
*
OR
*
EXAMPLE: Is caffeine dependence real?
Our subjects are 11 people diagnosed as being dependent of caffeine. Each subject was barred
from coffee, colas, and other substances containing caffeine. Instead they took capsules
containing their normal caffeine intake. During a different time period, they took placebo
capsules. The order in which subjects took caffeine and placebo was randomized. The table
below contains data on two of several tests given to the subjects. “Score” is the score on the
Beck Depression Inventory. Higher scores show more symptoms of depression. Construct and
interpret a 90% confidence interval for the mean change in depression score.
Subject
#
1
2
3
4
5
Score
(caffeine)
5
5
4
3
8
Score
(placebo)
16
23
5
7
14
Subject
#
6
7
8
9
10
11
SHOW ALL STEPS
Score
(caffeine)
5
0
0
2
11
1
Score
(placebo)
24
6
3
15
12
0
Estimating a Population Proportion (10.3)
Learning Targets:
p , determine the standard error of p .
Given a sample proportion,
List the 3 conditions that must be present before constructing a confidence
interval for an unknown proportion.
Construct a confidence interval for a population proportion, remembering to
use the 4-step procedure.
Determine the sample size necessary to construct a level C confidence
interval for a population proportion with a specified margin of error.
Conditions for Inference about a Proportion:
As always, inference is based on the sampling distribution of a statistic. We described the
sampling distribution of a sample proportion p in section 9.2. Here is a brief review of its
important properties:
Center:
The mean is ________. The sample proportion is an ___________________
estimator of the population proportion, p.
Shape:
The standard deviation of
p is ________________________, provided that
the population is at least ______ times as large as the sample.
Spread:
The distribution of
p is approximately Normal if the sample size is large enough
that ______________________ and _______________________.
In practice, we don’t know the value of p. (If we did, we wouldn’t need a confidence interval
for it). So we won’t be able to check the conditions for a Normal distribution given above. In
large samples, p will be ________________ to
p.
So we replace p by p in determining the
conditions for Normality. We also replace p with p in the formula for standard deviation.
Since we are using an estimated value, standard deviation is now called the
______________________________of p .
SE 

p 1p

n
We use this in the confidence interval formula:
estimate

margin of error
REVIEW:
Steps to displaying a confidence interval:
P:
A:
N:
I:
C:
EXAMPLE 1:
As part of a quality improvement program, your mail-order company is
studying the process of filling customer orders. Company standards say an order is shipped on
time if it is sent out within 3 working days after it is received. You audit an SRS of 100 of the
500 orders received in the past month. The audit reveals that 86 of these orders were
shipped on time. Find a 95% confidence interval for the true proportion of the month’s orders
that were shipped on time.
P:
A:
N:
I:
C:
EXAMPLE 2:
A national opinion poll found that 44% of all American adults agree that
parents should be given vouchers good for education at any public or private school of their
choice. The result was based on a small sample. How large of an SRS is required to obtain a
margin of error of 0.03 (that is,  3%) in a 95% confidence interval?
EXAMPLE 3:
A company has received complaints about its customer service. They intend
to hire a consultant to carry out a survey of customers. Before contacting the consultant, the
company president wants some idea of the sample size that she will be required to pay for.
One critical question is the degree of satisfaction with the company’s customer service a
person has, measured on a five-point scale. The president wants to estimate the proportion, p,
of customers who are satisfied (that is, who choose either “satisfied” or “very satisfied,” the
two highest levels on the 5-point scale.) She decides she wants the estimate to be within 2%
(0.02) at a 95% confidence level. Find the margin of error used to estimate the proportion of
customers who are satisfied.
Chapter 10: Estimating with Confidence
Key Vocabulary:
Statistical inference
Confidence levels
Critical values
t-distribution
z-distribution
Paired t procedures
Confidence intervals
Margin of Error
Standard error
Degrees of freedom
one-sample t interval
Robust
10.1 Confidence Intervals: The Basics
1. What does a confidence interval estimate?
2. Explain the difference between a confidence interval and a confidence level.
3. Describe the conditions that must be met in order to determine a confidence interval.
4. What is meant by a critical value?
5. Explain margin of error. What error does it cover?
6. List three situations in which the margin of error gets smaller.
10.2 Estimating a Population Mean
1. Why do we use standard error?
2. Under what assumptions is Sx a reasonable estimate of  ?
3. What is a t-distribution?
4. List 3 facts about the t-distribution.
5. Why do we use degrees of freedom?
6. Describe the difference between a one-sample t-interval and a two-sample t-interval
7. If a procedure is robust, it is not strongly affected by ______________________.
10.3 Estimating a Population Proportion
1. Explain how “center”, “shape”, and “spread” help determine the conditions/assumptions for
inference about a proportion.
2. In statistics, what is meant by a sample proportion?
3. What is the confidence interval estimating?
4. When is p approximately Normal?
5. How do you calculate the standard error of p ?
6. How do you determine a confidence interval for p?
Significance Tests – The Basics (11.1)
Learning Targets:
Explain why significance testing looks for evidence against a claim rather than in favor of
the claim.
Define Null Hypothesis and Alternative Hypothesis.
Explain the difference between a one-sided hypothesis and a two-sided hypothesis
Identify the three conditions that need to be present before doing a significance test for a
mean
Explain what is meant by a test statistic. Give the general form of a test statistic
Define P-value
Define significance level
Define statistical significance at level  .
Explain the difference between the P-value approach to significance testing and the
statistical significance approach.
Introduction:
In the last chapter, we found confidence intervals to _________________
___________________________________. The other common type of statistical
inference, called significance tests, has a different goal: to assess the evidence provided by
data about some claim concerning a population.
EXAMPLE:
I claim that I make 80% of my basketball free throws. To test my claim, you
ask me to shoot 20 free throws. I make only 8 of the 20. “Aha!” you say. “Someone who
makes 80% of their free throws would almost never make only 8 of the 20. So I don’t believe
your claim.”
Your reasoning is based on asking what would happen if my claim were true and we repeated
the sample of 20 free throws many times – I would almost never make as few as 8.This
outcome is so unlikely that it gives strong evidence that my claim is not true.
The Basics:
A significance test is a formal procedure for comparing __________________
with a _________________. The hypothesis is a statement about a population Parameter, like
the population mean,
 , or population proportion, p.
The reasoning of statistical tests is based
on asking what would happen if we repeated the sampling or experiment many times. As in the
previous chapter, we begin with the unrealistic assumption that we know the population standard
deviation,
.
The Hypotheses:
We will be writing a null hypothesis, which always states that
____________________________, and an alternative hypothesis, which suggests that
____________________________________________________. We abbreviate the null
hypothesis as H 0 and the alternative hypothesis as Ha . Hypotheses always refer to some
population, not a particular outcome. Be sure to state H 0 and Ha in terms of a population
parameter.
EXAMPLE:
Diet colas use artificial sweeteners to avoid sugar. Colas with artificial
sweeteners gradually lose their sweetness over time. Manufacturers therefore test new colas
for loss of sweetness before marketing them. Trained tasters sip the cola along with drinks of
standard sweetness and score the cola on a “sweetness score” of 1 to 10. The cola is then
stored for a month at high temperature to imitate the effect of 4 months’ storage at room
temperature. After a month, each taster scores the stored cola again. This is a matched-pairs
experiment. Our data are the differences in the tasters’ scores (score before storage minus
score after storage). The bigger these differences, the bigger the loss of sweetness.
Here are the sweetness scores for a new cola, measured by 10 trained tasters:
2.0
0.4
0.7
2.0
-0.4
2.2
-1.3
1.2
1.1
2.3
Most tasters found a loss of sweetness, but 2 found a gain in sweetness. Assume the standard
deviation is 1. We need to know if this data is good evidence that the cola actually lost
sweetness in storage. The sample mean x is calculated to be __________. The population
mean  would represent ___________________________________________________
* What are we saying if “there is no change” ? ___________________________________
* What does the evidence point to? ____________________________________________
H 0 : ________________
Ha : ________________ (One-sided or two-sided?)
EXAMPLES:
Practice stating hypotheses:
1. Suppose a dog food manufacturer wants to know if the proper amount of dog food is being
placed in the 25-lb bags. (One-sided alternative or two-sided alternative?)
H0 :
Ha :
2. The Standard Tire Company has introduced a new tire in Europe that will be guaranteed to
last at least 30,000 km. An independent agency conducted several tests and suspects the tires
do not last as long as claimed. (One-sided alternative or two-sided alternative?)
H0 :
Ha :
The Assumptions:
In chapter 10, we had three conditions that should be satisfied before
we construct a confidence interval about an unknown population mean or proportion. These
same 3 conditions must be verified before performing significance tests about a population
mean or proportion:
* _________________
* _____________________
* __________________
As in the previous chapter the details for checking the Normality condition are different for
means and proportions.
For means: ________________________________________________________
For proportions: ____________________________________________________
The Test Statistics:
The hypothesis test is based on a statistic comparing the value of
the parameter as stated in the null hypothesis with an estimate of the parameter from the
sample data. If the estimate is far from the parameter, we have evidence against the null
hypothesis. To assess how far the estimate is from the parameter, we standardize the
estimate:
EXAMPLE:
Find the test statistic for the Diet Cola problem.
The P-Values:
The null hypothesis states the claim we are seeking evidence against. The
test statistic measures how much the sample data diverge from the null hypothesis. The
amount of divergence tells us we have data that would be unlikely if H 0 were true. “Unlikely”
is determined by a probability, called a P-value. We compute the probability assuming
_____________________________________________. The P-value tells us how likely it
is that the sample data would occur when we assume the null hypothesis is true. (Maybe we
just drew a bad sample, but another sample would be closer to the parameter). Small P-values
are evidence against H 0 because they say that the observed result is unlikely to occur when
H0 is true (Probably not just a bad sample). Once we find the test statistic, we find the
probability of this value occurring under the assumed H 0 .
EXAMPLE:
Find the P-value for the Diet Cola problem.
If the P-value (probability) is small enough we reject the claim that the null hypothesis is
making. How small is “small enough”?
The Significance Levels:
We must have some value to compare the P-value to. It is
very important to understand that this value, called the significance level (  - value), must be
chosen during the design stage of the experiment, not after the data is collected. The
“significance” refers to how likely it is for something to happen. For example, does it have a
5% chance of occurring? Does it have a 1% chance of occurring? If the P-value is less than
the  -value, we say ___________________________________________, meaning they
are significant enough to reject the claim of the null hypothesis. When deciding on an  value, determine how important the decision is. Life-threatening situations usually require a
0.01 significance level, while the height of soap suds would require a 10% significance level.
EXAMPLE:
Assume in the Diet Cola problem, the significance level (  ) is 5%. That is  =
0.05 Compare the P-value found in the previous example, and decide if the results are
significant.
There is a difference between “statistical significance” and “practical significance”. We will
get to that later in the chapter.
The Conclusion:
The final step in performing a significance test is to draw a conclusion
about the null hypothesis: “reject” or “do not reject”, we NEVER “ACCEPT”. Your conclusion
should have a clear connection to your calculations and should be stated in the context of the
problem. We reject H 0 if our sample result is too unlikely to have occurred by chance
assuming H 0 is true.
EXAMPLE:
Write a conclusion for the Diet Cola problem.
To help remember the steps in a significance test, you might use this phrase:
Parameter
(What parameter are we using? What does it represent?)
Hypotheses
(null and alternative)
Assumptions
(SRS, Normality, Independence)
Name the test
(one-sample z-test… for now …)
Test statistic
(show the formula used and give the value)
Obtain p-value
(finding probability)
Make decision
(“reject” or “do not reject”)
State conclusion
(decision made in the step above, what made you decide
that, what it means in context of the problem)
Carrying Out Significance Tests (11.2)
Learning Targets:
Identify and explain the steps involved in formal hypothesis testing
Conduct a z-test for a population mean
Explain the relationship between a level  two-sided significance test for
 and a level 1   confidence interval for  .
Conduct a 2-sided significance test for  using a confidence interval
Results of a significance test hold up in court because it points to a result that is unlikely to occur
simply by chance.
Two-sided tests:
Two-sided, or two-tailed, tests occur when our alternative hypothesis is twosided. We need to check both tails of the Normal curve when finding the P-value of a two-tailed test.
EXAMPLE:
The medical director of a large company is concerned
about the effects of stress on the company’s younger executives.
According to the National Center for Health Statistics, the mean
systolic blood pressure for males 35 to 44 years old is 128, and the
standard deviation in this population is 15. The medical director
examines the medical records of 72 male executives in this age group
and finds that their mean systolic blood pressure is x  129.93 . Is this
evidence that the mean blood pressure for all the company’s younger
male executives is different from the national average?
Tests from Confidence Intervals:
A 95% confidence interval captures the
_____________________________________________ in 95% of all samples. If we are
95% confident of this idea, we are also confident that values outside of our confidence
interval are incompatible with our data. There is a __________ chance that we won’t capture
the true value of the mean. So A 5% significance level and a 95% confidence interval can be
used to draw the same conclusions. We can say the same about a _________ confidence
interval and a _________ significance level.
EXAMPLE:
The Deely Laboratory analyzes specimens of a drug to determine the
concentration of the active ingredient. Such chemical analyses are not perfectly precise.
Repeated measurements on the same specimen will give slightly different results. The results
of repeated measurements follow a Normal distribution quite closely. The analysis procedure
has no bias, so the mean  of the population of all measurements is the true concentration of
the specimen. The standard deviation of this distribution is a property of the analysis method
and is known to be  = 0.0068 grams per liter. The laboratory analyzes each specimen three
times and reports the mean result.
A client sends a specimen for which the concentration of active ingredient is supposed
to be 0.86%. Deely’s three analyses give concentrations 0.8403, 0.8363, 0.8447. Give a 99%
confidence interval for the concentration of the active ingredient. Is this significant evidence
(at the 1%) level that the true concentration is not 0.86%?
Use and Abuse of Tests (11.3)
Learning
Targets:
Distinguish between statistical significance and practical importance
Identify the advantages and disadvantages of using P-values rather than a
fixed level of significance
Introduction:
Significance tests are frequently used when reporting results of research in
many fields. New drugs require significant evidence of effectiveness and safety. Courts ask
about statistical significance in hearing discrimination cases. Marketers want to know whether
a new ad campaign significantly outperforms the old one, and medical researchers want to know
whether a new treatment performs significantly better.
Choosing a level of significance:
When designing a significance test, you should choose  before performing the test. Consider
how much evidence is required to reject H0.
*
How plausible is H0? - If H0 represents an assumption that the people you must convince
have believed for years, it’s going to take strong evidence (small  ) to convince them.
*
What are the consequences of rejecting H0? - If rejecting H0 means an expensive change,
you will need strong evidence.
* 
is not a hard-core comparison value. - There is not a practical distinction between
P = 0.049 and P = 0.051.
Statistical Significance and Practical Importance:
When large samples are available, even tiny deviations from the null hypothesis will be
significant. Always remember to check the practical significance.
EXAMPLE:
Suppose a dog food manufacturer wants to know if the proper amount of dog
food is being placed in the 25-lb bags. Suppose an SRS of 2000 bags was selected, and
x  25.01 pounds with   0.1 pound. Is this enough evidence that the bag-filling equipment
needs to be adjusted? Show all steps
Statistical inference is not valid for all sets of data:
Badly designed surveys or experiments often provide invalid results. Our tests of significance
cannot correct flaws in the design. Don’t be too impressed by P-values on a printout until you
are confident that the data was produced correctly.
EXAMPLE:
You wonder whether background music would improve the productivity of the
staff who process mail orders in your business. After discussing the idea with the workers, you
add music and find a statistically significant increase. Should you conclude improvement due to
background music?
Change in environment
Study under way
What needs to be added?
Don’t ignore lack of significance:
There is a tendency to infer “no change” or “no effect” whenever the P-value fails to attain the
usual 5% standard. Remember, we simply “fail to reject H0” when the P-value is not less than  .
That does not mean that H0 is true. Maybe we need to increase our sample size and perform the
significance test again.
Using Inference to Make Decisions (11.4)
Learning Targets: Define what is meant by a Type I error and a Type II error
Introduction:
Describe, given a real situation, what constitutes a Type I error and what the consequences
of such an error would be
Describe, given a real situation, what constitutes a Type II error and what the consequences
of such an error would be
Describe the relationship between significance level and a Type I error
Define what is meant by the power of a test
Identify the relationship between the power of a test and a Type II error
List 4 ways to increase the power of a test
Explain why a large value for the power of a test is desirable.
Tests of significance assess the _____________________________________________.
We measure evidence by the P-value, which is “probability, computed under the assumption
that ____________________________________________”. The alternative hypothesis
enters the test only to ____________________________________________________.
When we choose  before performing the test, we are using the outcome of the test to
______________________________.
EXAMPLE:
A producer of bearings and the consumer of the bearings agree that each
carload lot must meet certain quality standards. When a carload arrives, the consumer
inspects a sample of the bearings. On the basis of the sample outcome, the consumer either
accepts or rejects the carload. This is called ____________________________________.
For this situation, we must decide between
H0: the lot of bearings meets standards
Ha: the lot of bearings does not meet standards
We have 4 possible situations, here:
A)
B)
C)
D)
What is the difference between the two types of errors?
Whether or not we make one of these errors depends on the performance of the significance
test. So the probability of making one of these errors will describe the performance of the
test.
Probability of Making a Type I Error:
EXAMPLE:
The mean diameter of a type of bearing is supposed to be 2.000cm. The
bearing diameters vary normally with standard deviation  =0.010cm. When a carload lot of
the bearings arrive, the consumer takes an SRS of 5 bearings from the lot and measures their
diameters. The consumer rejects the bearings if the sample mean diameter is significantly
different from 2cm at the 5% significance level.
Probability of Making a Type II Error:
Step 1: ________________________________________________________________
Step 2:_______________________________
Step 3: ________________________________________________________________
(to standardize, you will be given a value to use as
a )
EXAMPLE:
Find the probability of making a Type II error in the bearing problem. Use a
significance level of 5%, and  a = 2.015:
A test makes a type II error when it fails to reject a null hypothesis that really is false. A
high probability of a type II error means that __________________________________.
We usually report the sensitivity of the test as the strength, or the ________________, of
the test. This is the probability that the test will reject H0 when it’s supposed to reject it
(when Ha is actually true). To calculate, use POWER = 1 – P(type II error)
** Calculations of P-values and calculations of power both say ___________________
____________________________________________________________.
** P-value describes _________________________________________________.
** Power tells us ____________________________________________________.
EXAMPLE:
The cola maker in the “loss of sweetness” problem determines that a sweetness loss
is too large to accept if the mean response for all tasters is  a = 1.1. Will a 5% significance test of
the hypotheses:
H0 :   0
Ha :   0
based on a sample of 10 tasters usually detect a change this great?
(Previously,
 = 1)
Chapter 11: Testing a Claim
Key Vocabulary:
Null hypothesis
P-value
Test statistic
Significance level
Test of significance
Type I error
Alternative hypothesis
Statistically significant
Practical importance
Upper p critical value
Two-sided test
Type II error
11.1 Significance Tests: The Basics
1. What is a null hypothesis?
2. What is an alternative hypothesis?
3. What is a test statistic? How do you find it?
4. What is meant by a P-value?
5. How does a P-value give information about the significance test?
6. What does the alpha-value do in a significance test?
11.2 Carrying Out Significance Tests
11.3 Use and Abuse of Tests
1. Explain the 3 conditions necessary to perform inference
2. Explain the difference between accepting H0 and failing to reject H0.
3. Explain how we can use a confidence interval to make a decision about H0.
4. What should be considered when deciding on a level of significance?
5. Explain the purpose of example 11.15
11.4 Using Inference to Make a Decision
1. Explain the difference between a Type I error and a Type II error.
2. What is the relationship between the significance level and the probability of a
Type I error?
3. Describe briefly how to find the power of a significance test.
4. What does the power of a significance test tell us?
5. List 4 ways you can increase the power of a significance test.
Inference for the Mean of a Population (12.1)
Learning Targets:
Define the one-sample t-statistic
Determine critical values of t (t*), from a “t-table” given the probability of
being to the right or left of t*
Determine the P-value of a t-statistic for both a one-sided and two-sided
significance test
Conduct a one-sample t significance test for a population mean using the
required steps
Conduct a paired t-test for the difference between two population means
Introduction:
Now that we have studied the principles and process of significance tests,
we move to the practical application. We begin by dropping the unrealistic assumption that we
know the population standard deviation,  . We use the t-distribution, as we did in chapter 10
with confidence intervals. Let’s review some of the characteristics of the t-distribution:
* Since  was replaced by s, the statistic has more ________________ and no longer has a
normal distribution.
* We specify a particular t-distribution by giving its ________________________.
* The density curves of the t-distributions are similar in shape to the _____________
______________________________. They are symmetric about ________, singlepeaked, and ________________________.
* The spread of the t-distributions is a bit _________________ than that of the
standard normal distribution. The t-distributions have more area/probability in
___________________ and less in the __________________ than the standard normal
distribution does.
* As the degrees of freedom, k, increase, the t(k) density curve approaches the
_____________________________more closely.
The t-distribution:
Since the population standard deviation is unknown, and must be
estimated with the ____________________________________, we cannot find the
standardized value, z to use as the test statistic. There is another standardized value we can
use, called ______. The formula is very similar to the z-score formula:
Like z, this standardized value
t
tells us how many standard-deviation units
Unlike z, there is a different t-distribution for each sample size.
A significance test using the test statistic t is called a one-sample t-test.
x is from  .
Assumptions:
One-sample t-procedures are exactly correct when the population is
_____________, which never happens in reality. We assume that the population is
_____________________________ in order to justify the use of t-procedures. The tprocedures are strongly influenced by ____________. The results will not be reliable if
there are _______________ and the sample size is ________________. For small sample
sizes, sizes of less than ______, only use t-procedures if the data are close to normal. If the
sample size is at least _______, only use t-procedures if there is no
____________________. We can safely use t-procedures if the sample size is at least
________, even if the data is skewed. Outliers in any sample size make the t-procedures
invalid.
YOU MUST MAKE A BOX-POLT ON YOUR CALCULATOR AND CHECK ITS SYMMETRY BEFORE
ASSUMING THE DATA IS NEARLY NORMAL. COMMENTS ABOUT THE SHAPE OF THE BOX
PLOT ARE REQUIRED IN THE ASSUMPTIONS STEP.
EXAMPLE:
Many homeowners buy detectors to check for the invisible gas, radon, in their
homes. How accurate are these detectors? To answer this question, University researchers
placed 12 radon detectors in a chamber that exposed them to 105 picocuries per liter of radon.
The detector readings were as follows:
91.9
97.8
111.4
122.3 105.4 95.0
103.8 99.6
96.6
119.3 104.8 101.7
a) Make a boxplot of the data. Describe the shape of the distribution.
b) Is there convincing evidence that the mean reading of all detectors of this type differs
from the true value of 105? Carry out a test in detail, and then write a brief conclusion.
(Use the tcdf function on the calculator during the “O” step.)
P:
H:
A:
N:
T:
O:
M:
S:
Paired t-tests:
Remember the diet cola “loss of sweetness” problem? The same 10
tasters rated before and after sweetness. We subtracted the before sweetness score and
the after sweetness score and performed a test on the differences. This is the method of a
paired t-test.
EXAMPLE:
The acculturation Rating Scale for Mexican Americans (ARSMA) and the
Bicultural Inventory (BI) both measure the extent to which Mexican Americans have adopted
Anglo/English culture. These tests were compared by administering both tests to 22 Mexican
Americans. Both tests have the same range of scores (1.00 to 5.00) and are scaled to have
similar means for the groups used to develop them. There was a high correlation between the
two scores, giving evidence that both are measuring the same characteristics. The
researchers wanted to know whether the population mean difference in scores for the two
tests is 0. The differences in scores (ARSMA – BI) for the 22 participants had x  0.2519
and s = 0.2767. Find the test statistic and the P-value to answer the researchers’ question.
P:
H:
A:
N:
T:
O:
M:
S:
Tests about a Population Proportion (12.2)
Learning Targets:
Explain why p 0 instead of p is used when computing the standard error
of p in a significance test for a population proportion.
Explain why the correspondence between a two-tailed significance
test and a confidence interval for a population proportion is not as
exact as when testing for a population mean.
Explain why the test for a population proportion is sometimes called a
large sample test.
Conduct a significance test for a population proportion using the
PHANTOMS steps.
Discuss how significance tests and intervals can be used together to
help draw conclusions about a population proportion.
Introduction:
The proportion of a population having a given characteristic is a parameter
called ______. The proportion of a sample having a given characteristic is a statistic called
__________. The problems in this section are the same type as in chapter 11 and 12.1, where
we are hypothesizing about a _________________, but instead of  , we are making
inferences about _______. Instead of x , we use _____ to standardize and find the P-value.
We still do not know the population standard deviation, ______. Instead of using s, we
calculate an estimate (standard error) using a formula similar to the one used in confidence
intervals:
Assumptions:
For sufficiently large samples, we know that the sampling distribution of p
is approximately _______________ (provided p is not too close to _______ or _______),
with a mean equal to _____ and standard deviation equal to ______________. To
standardize p , we use the z-score formula:
Assumptions for inference about a proportion
*
*
*
(one-proportion z-test):
Hypothesis Tests:
H0
Ha
Find z:
Confidence Intervals:
Find p-value:
The formula for finding a confidence interval is:
The interval is describing ________________________________________________
How does the confidence interval support a decision made in a significance test?
EXAMPLE 1:
As part of a quality improvement program, your mail-order company is
studying the process of filling customer orders. Company standards say an order is shipped on
time if it is sent out within 3 working days after it is received. You audit an SRS of 100 of the
500 orders received in the past month. The audit reveals that 86 of these orders were
shipped on time.
Find a 95% confidence interval for the true proportion of the month’s orders that were
shipped on time.
P
A
N
I
C
Is there good evidence that the proportion of orders shipped on time differs from last
month’s average of 88%?
EXAMPLE 2:
According to the National Institute for Occupational Safety and Health, job
stress poses a major threat to the health of workers. A national survey of restaurant
employees found that 75% said that work stress had a negative impact on their personal lives.
A random sample of 100 employees from a large restaurant chain finds that 68 answer “yes”
when asked, “Does work stress have a negative impact on your personal life?” Is this good
reason to think that the proportion of all employees in this chain who would say “yes” differs
from the national proportion p0 = 0.75?
P
H
A
N
T
O
M
S
Chapter 12: Significance Tests in Practice
Key Vocabulary:
Standard Error
Degrees of Freedom
Paired t-tests
t Distribution
One-sample t-statistic
One-prop z-test
12.1 Tests about a Population Mean
1. Who does the author say developed the t-distributions?
2. What was he trying to do when he noticed the need for t-distributions?
3. What is the interpretation for the t-statistic?
4. Describe the differences between a standard normal distribution and a t distribution.
5. Explain how normality is checked for a population mean.
6. How do you calculate the degrees of freedom for the t-distribution?
7. What happens to the t-distribution as the degrees of freedom increase?
8. In a matched pairs t-procedure, what is the parameter of interest,  ?
9. What measures a tests ability to detect deviations from the null hypothesis?
10. How would you construct a level C confidence interval for
11. Under what assumptions is s a reasonable estimate of


if

is unknown?
?
12.2 Tests about a Population Proportion:
1. Explain how normality is checked for a population proportion.
2. Why do some people call this a “large-sample test”?
3. Why are confidence intervals sometimes included as part of the analysis?
4. Give the formula for standard error used in finding a confidence interval.
5. Give the formula for standard error used in a one-proportion z-test.
6. Explain why we use different formulas in the confidence interval and the test statistic?
Comparing Two Means (13.1)
Learning Targets: Identify situations in which two-sample problems might arise
Describe the three conditions necessary for doing inference involving two population
means
Clarify the difference between the two-sample z-statistic and the two-sample tstatistic
Identify the two practical options for using two-sample t-procedures and how they
differ in terms of computing the number of degrees of freedom
Conduct a 2-sample significance test for the difference between two-independent
means using PHANTOMS
Compare the robustness of two-sample procedures with that of one-sample
procedures. Include in your comparison the role of equal sample sizes
Explain what is meant by “pooled two-sample t – procedures,” when pooling can be
justified, and why it is advisable not to pool
Introduction:
Comparative studies are more convincing than single-sample investigations, so _________
inference is not as common as comparative __________________________________. In a
comparative study, we may want to compare two ____________, or we want to compare two
_______________. In either case, the samples must be chosen ______________ and
____________________ in order to perform statistical inference. Because matched pairs
are not chosen independently, we will not use two-sample inference for a matched pairs design.
Instead, we apply the one-sample __________________to the observed
__________________.
If the same subjects are creating both data sets, it is a matched pairs design.
EXAMPLE 1
The following situations require inference about a mean or means. Identify each as a singlesample, a matched-pairs sample, or a two-sample:
A) An education researcher wants to learn whether it is more effective to put questions
before or after introducing a new concept in an elementary school mathematics text. He
prepares two textbook segments that teach the same concept, one with motivating questions
before and the other with review questions after. He uses each text segment to teach a
separate group of children. The researcher compares the scores of the groups on a test over
the material.
B) Another researcher approaches the same issue differently. She prepares textbook
segments on two unrelated topics. Each segment comes in two versions, one with questions
before and the other with questions after. The subjects are a single group of children. Each
child studies both topics, one (chosen at random) with questions before and the other with
questions after. The researcher compares test scores for each child on the two topics to see
which topic he or she learned better.
Comparing Two Means
The null hypothesis for a two-sample hypothesis test still says there is no difference, but now
there are two  ‘s. You can use: ______________ or ___________________
The alternative hypothesis could be:
_______________________ or _______________________(two-sided)
_______________________ or _______________________(one-sided)
_______________________ or _______________________(one-sided)
Before you begin, check your assumptions:
If these assumptions hold, then the difference in sample means is an unbiased estimator of the
difference in population means, which tells us that ______________________.
Furthermore, if both populations are normally distributed, then __________ is also normally
distributed. Also, the variance of x1  x2 is the sum of the variances of _____ and ______.
So the standard error for the two-sample means is:
The two-sample t-statistic formula is similar to the one-sample t-statistic formula:
The degrees of freedom is taken to be the smaller of n1 - 1 and n2 - 1
EXAMPLE 2
In a study of cereal leaf beetle damage on oats, researchers measured the number of beetle
larvae per stem in small plots of oats after randomly applying one of two treatments: no
pesticide or Malathion at the rate of 0.25 pound per acre. The data appear roughly normal.
Here are the summary statistics:
Group
Treatment
n
x
s
1
2
control
Malathion
13
14
3.47
1.36
1.21
0.52
Is there significant evidence at the 1% level that Malathion reduces the mean number of larvae
per stem?
P
H
A
N
T
O
M
S
Estimating Two Means with Confidence
We can also find confidence intervals with two-sample t-procedures. The formula is similar to
the one-sample t-procedure:
The confidence interval says: ________________________________________________
or _____________________________________________________________________
EXAMPLE 3
Ordinary corn doesn’t have as much of the amino acid “lysine” as animals need in their feed.
Plant scientists have developed varieties of corn that have increased amounts of lysine. In a
test of the quality of high-lysine corn as animal feed, an experimental group of 20 one-day-old
male chicks received a ration containing the new corn. A control group of 20 one-day-old male
chicks received a ration that was identical except it contained normal corn. Here are the
weight gains (in grams) after 21 days:
CONTROL
380
283
356
350
345
321
349
410
384
455
366
402
329
316
360
EXPERIMENTAL
356
462
399
272
431
361
434
406
427
430
447
403
318
420
339
401
393
467
477
410
375
426
407
392
326
a) Check your assumptions for comparing two means
b) Is there good evidence that chicks fed high-lysine corn gain weight faster?
c) Give a 95% confidence interval for the mean extra weight gain in chicks fed high-lysine
corn.
Robustness
Recall from chapter 10, an inference procedure is called robust if the probability calculations
involved in the procedure remain fairly accurate when a condition for use is violated. The
_____________________ procedures are more robust than the ____________________
procedures, particularly when the distributions are not symmetric. When the sizes of the two
samples are _________________ and the two populations being compared have distributions
with similar shapes, probability values from the t-table are quite accurate. A broad range of
distributions is included when the sample sizes are as small as n1  n2  5 . When the two
distributions have different _______________________, larger samples are needed.
As a guide, we will adapt the sample size conditions from the one-sample t-procedures by
replacing “sample size” with “the sum of the sample sizes” as long as n1 and n2 are both at
least 5.
* Except in the case of small samples, the assumption that the data are an SRS from
the
population of interest is more important than the assumption that the
population distribution is Normal
*
Sum of the sample sizes is less than 15 : Use t-procedures only if the data are
close to Normal.
*
Sum of the sample sizes is at least 15 : The t-procedures can be used except in
the
presence of _________________________ or strong
_________________________
*
Large samples : The t-procedures can be used even for clearly skewed distributions
when the sum of the samples is large (n > 30).
WE CAN NEVER USE T-PROCEDURES WHEN OUTLIERS ARE PRESENT
In planning a two-sample study, choose equal sample sizes if you can. The two-sample tprocedures are most robust against non-Normality in this case, and the conservative P-values
are most accurate. We can, however, complete a study even if the sample sizes are not equal.
The Pooled Two-Sample t-Procedures
(Don’t Use Them!)
Pooled two-sample t-procedures average the two sample variances to estimate the population
standard deviation. It is only equal to our t-statistic if the two sample sizes are the same, but
not otherwise.
Comparing Two Proportions (13.2)
Learning Targets:
Identify the mean and standard deviation of the sampling distribution of
p1  p2
List the conditions under which the sampling distribution of p1  p2 is
approximately Normal
Identify the standard error of p1  p2 when constructing a confidence
interval for the difference between two population proportions
Construct a confidence interval for the difference between two population
proportions using the PANIC method for confidence intervals
Explain why, in a significance test for the difference between two
proportions, it is
reasonable to combine (pool) your sample estimate
to make a single estimate of the difference between the proportions.
Explain how the standard error of p1  p2 differs between constructing a confidence interval for p1  p 2
and performing a hypothesis test for H0 : p1  p2  0
List three conditions that need to be satisfied in order to a significance test for the difference
between two proportions
Conduct a significance test for the difference between two proportions using PHANTOMS
Introduction:
In section 13.1, we discussed a need to compare two ____________________ or two
__________________________. We used a ____________________________. We can
still make these comparisons when using proportions. We call it a _____________________
_________________________________.
Two-Sample Problems: Proportions
We will use notation similar to that used in our study of two-sample t statistics. The groups
we want to compare are Population 1 (or Treatment 1) and Population 2 (or Treatment 2). We
have a separate SRS from each population (or treatment). Here is our notation:
Population
population proportion
sample size
sample
proportion
1
p1
n1
p1
2
p2
n2
p2
We compare the populations by doing inference about the difference p1  p2 . The statistic
that estimates this difference is the difference between the two sample proportions p1  p2 .
Significance Tests for Comparing Two Proportions:
An observed difference between two sample proportions can reflect a difference in the
populations, or it may just be due to chance variation in random sampling. Significance tests
help us decide if the effect we see in the samples is really there in the populations. The null
hypothesis says there is _______________________ between the two parameters.
The alternative hypothesis says what kind of difference we expect:
_______________________ or _______________________(two-sided)
_______________________ or _______________________(one-sided)
_______________________ or _______________________(one-sided)
Before you begin, check your assumptions:
*
*
*
To do a test, standardize p1  p2 to get a z-statistic. If H0 is true, all the observations in
both samples really come from a single population with a single unknown proportion. So
instead of estimating p1 and p2 separately, we combine the two samples and use the overall
sample proportion to estimate the single population parameter, pc. Call this the combined
sample proportion (or the “pooled” sample proportion). It is:
Use p c in place of both p1 and p 2 in the expression for the standard error:
Using this standard error will give us a z-statistic that has the standard Normal distribution
when H0 is true. The formula for the z-score is:
EXAMPLE:
The 1958 Detroit Area Study was an important investigation of the influence of
religion on everyday life. The sample “was basically a simple random sample of the population
of the metropolitan area” of Detroit, Michigan. Of the 656 respondents, 267 were white
Protestants and 230 were white Catholics. The study took place at the height of the cold war.
One question asked whether the government was doing enough in areas such as housing,
unemployment, and education. 161 of the Protestants and 136 of the Catholics said no. Is
there enough evidence that the 267 white Protestants and the 230 white Catholics differed
on this issue?
P:
H:
A:
N:
T:
O:
M:
S:
Confidence Interval for Comparing Two Proportions:
Confidence Interval = Estimate  Margin of Error
= Estimate  z* (Standard Error)
=
Conditions:
*
*
*
EXAMPLE:
Another question from the 1958 Detroit Area study asked if the right of free
speech included the right to make speeches in favor of communism. Of the 267 white
Protestants, 104 said yes, while 75 out of the 230 white Catholics said yes. Check that it is
safe to use the z-confidence interval, then give a 95% confidence interval for the difference
between the proportion of Protestants who agreed and the proportion of Catholics who agreed.
Chapter 13: Comparing Two Population Parameters
Vocabulary: Two-sample problems
Two-sample z statistic
Robust
Combined sample proportion
Two-sample t statistic
Pooled variances
13.1 Comparing Two Means
1. How are two-sample problems different than one-sample problems?
2. How are the conditions for two-sample procedures different from the conditions for onesample procedures?
3. Here are the notations used in two-sample procedures. Complete the table:
__________POPULATION
Population
Variable
1
x1
2
x2
_____
Mean
STATISTIC__________
St. Dev.
Sample size
Mean
n1
St. Dev.
s1
2
4. Explain when to use two-sample z inference and when to use two-sample t-inference.
5. Write the formula for the test statistic when using two-sample z-procedures.
6. Write the formula for the test statistic when using two-sample t-procedures.
7. How do we adjust the one-sample t guideline about sample sizes to work in the two-sample t
test?
13.2 Comparing Two Proportions
1.
Explain the difference between a matched-pairs study and a two-sample study.
2.
Explain why we can use the combined sample proportion, p C , in hypothesis tests but not
in confidence intervals.
3.
Explain how to calculate the standard error of p1  p2 .
4.
What assumptions must be met in order to use z-procedures for inference about two
proportions?
5.
Describe how to construct a level C confidence interval for the difference between two
proportions
p1 – p2.
6.
For a two-sample hypothesis test where H0: p1 = p2, what is the formula for calculating
the z-test statistic?
Test for Goodness of Fit (14.1)
Learning Targets:
Describe the situation for which the chi-square test for goodness of fit is
appropriate
Define the X2 statistic, and identify the number of degrees of freedom it is based on,
for the
 2 goodness of fit test
List the conditions that need to be satisfied in order to conduct a
 2 test
for goodness of fit
Conduct a
 2 test for goodness of fit
Identify three main properties of the chi-square density curve
Use technology to conduct a
 2 test for goodness of fit
If a X2 statistic turns out to be significant, discuss how to determine which observations contribute the
most to the total value
Introduction:
On average, the new mix of colors of M&M’s Milk Chocolate Candies will
contain 13% of each of browns and reds, 14% yellows, 16% greens, 20% oranges and 24% blues.
Assume you have a 1.69-ounce bag of M&M’s. After opening, you discover the following
candies:
Browns:
7
Reds:
8
Yellows:
9
Greens:
10
Oranges:
13
Blues:
13
Does your bag (sample) reflect the distribution advertised by the M&M/Mars Company? If so,
there should be very little difference between the ______________________ counts and
the ____________________ counts. But how much difference is too much?
The chi-square (“kie-square”) test for goodness of fit allows us to determine whether a
specified population distribution seems valid. To analyze categorical data, we construct one-way
tables and examine the counts in the __________________ for the explanatory and
_________________ variables. We then compare the observed counts to the expected counts.
One-Way Table:
COLOR
OBSERVED
COUNT
EXPECTED
COUNT
 observed  exp ected
2
exp ected
E
Blue
Brown
Green
Orange
Red
Yellow
TOTAL
Sum:  
2

O  E
E

O  E
2
=
2
The sum of the last column is called the ___________________________________. It
measures how well the __________________ counts fit the __________________ counts,
assuming that the null hypothesis is true. The larger this value is, the more evidence there will
be against H0.
P :
H0 :
Ha :
Properties of the Chi-Square Distribution:
* like the t-statistic, there are many chi-square distributions in the family. To choose
appropriate one, use the ________________________________________.
* the degrees of freedom (d.f.) is determined by the number of cells (rows) in the
one-way table. (d.f. = # rows – 1)
* all individual expected counts must be at least 1
* no more than 20% of the expected counts can be less than 5
(to ensure a sufficient sample size)
* the distribution is a density curve, beginning at ________ on the horizontal axis and
is skewed _______________.
* as the degrees of freedom increase, the shape of the curve becomes more _________.
*  2 pdf (x, d.f.) will graph the chi-square distribution
*
2 cdf
(lower, upper, d.f.) will calculate the area under the chi-square curve. Use
this to find the _____________ for the hypothesis test.
EXAMPLES:
1. Find the area to the right of X2 = 1.41 under the chi-square curve with 2 degrees of freedom.
2. Find the area to the right of X2 = 19.62 under the chi-square curve with 9 degrees of freedom.
3. A “wheel of fortune” at a carnival is divided into four equal parts:
part I:
Win a doll
part II:
Win a candy bar
part III:
Win a free ride
part IV:
Win nothing
You suspect that the wheel is unbalanced (i.e., not all parts of the wheel are
equally likely to be landed upon when the wheel is spun). The results of 500
spins of the wheel are as follows:
Part
Frequency
I
95
II
105
III
135
IV
165
Perform a goodness of fit test. Is there evidence that the wheel is not in balance?
P
H
A
N
T
O
M
S
I
II
III
IV
Inference for Two-Way Tables (14.2)
Learning Targets:
Explain what is meant by a two-way table.
Given a two-way table, compute the row or column conditional distributions
Define the chi-square (X2) statistic
Using the words “populations” and “categorical variables”, describe the major
difference between homogeneity of populations and independence
Identify the form of the null hypothesis in a
 2 test for homogeneity of populations
Identify the form of the null hypothesis in a
 2 test of association/independence
Given a two-way table of observed counts, calculate the expected counts for each cell
List the conditions necessary to conduct a
Use technology to conduct a
 2 test of significance for a two-way table
 2 test of significance for a two-way table
Introduction: In both sections of chapter 14, we are interested in comparing a set of
____________________ to a set of _______________________. In a goodness-of-fit
test, there is a single categorical variable that takes on values over a single population. But
what if we want to compare more than two groups? We may want to compare gender and
opinion on abortion, or background music and wine selection. We need a new way to present the
data and a new test. We will use a two-way, or contingency, table to present the data.
We studied these tables in chapter 4 as an organizational method for data analysis. Now, we
want to use the data to make decisions. When comparing multiple variables, we usually have
one of the following questions:
1. Do the data come from the same population, or do the populations differ?
2. Are the variables associated, or are they independent?
These questions determine which new Chi-square test to use:


EXAMPLE 1:
Suppose that data are collected on gender (M, F) and political party
preference (Republican, Democrat, Other). Suppose the data look like this:
Male
Female
Republican
24
2
Democrat
2
24
Other
1
1
Does there appear to be any relationship? _______________________________________
Are the variables independent? _____________
Suppose the data look like this:
Republican
24
24
Male
Female
Democrat
2
2
Other
1
1
Does there appear to be any relationship? _________________________________________
Are the variables independent? _____________
Finally, suppose the data look like this:
Republican
14
12
Male
Female
Democrat
11
10
Other
2
5
Does there appear to be any relationship? _________________________________________
Are the variables independent? _____________
Part 1 - Homogeneity of Populations:
Homogeneity (homogeneous):
Homogeneity of populations:
Statistical methods for dealing with multiple comparisons usually have two parts:
1. An overall test to see if there is good evidence of any differences among the
parameters that we want to compare.
2. A detailed follow-up analysis to decide which of the parameters differ and to
estimate how large the differences are.
EXAMPLE 2:
In a study of the television viewing habits of children, a developmental
psychologist selects a random sample of 300 first graders - 100 boys and 200 girls. Each child
is asked which of the following TV programs they like best: The Lone Ranger, Sesame Street,
or The Simpsons. Results are shown in the contingency table below.
Gender
Boys
Girls
Total
Lone Ranger
50
50
100
TV Show
Sesame Street
30
80
110
The Simpsons
20
70
90
Total
100
200
300
Do the boys' preferences for these TV programs differ significantly from the girls'
preferences?
The Problem of Multiple Comparisons:
The researchers expected that gender would influence choice of TV show, so gender is the
________________ variable, and the favorite TV show is the __________________
variable.
If we used a chi-square goodness of fit, we would have to do it three times:
H0: the distribution for gender and Lone Ranger is the same as
the distribution for gender and Sesame Street
H0: the distribution for gender and Lone Ranger is the same as
the distribution for gender and The Simpsons
H0: the distribution for gender and The Simpsons is the same as
the distribution for gender and Sesame Street
The problem is _______________________________________________. We need to be
able to do many comparisons at once with some overall measure of confidence.
For this type of problem, we are looking at a situation in which there are separate categorical
variables taking values on separate _____________________. We will use a 2-way table to
organize our data. It is similar to a matrix with “r” rows and “c” columns, so we call it an r x c
table. The test we perform looks at whether each treatment (TV show) affected the
populations (gender) differently or not is called the Chi-Square Test for Homogeneity of
Populations.
Stating Hypotheses:
We will still use a null hypothesis that says ______________________________________,
and an alternative hypothesis that says _________________________________________.
Identify the populations involved:
In this setting, the null hypothesis becomes:
The alternative hypothesis would be:
Computing the Expected Cell Counts:
We will still use the ________________ test to measure how far the observed values are
from the expected values. In this type of problem, however, we will also need to calculate the
______________.
expected count =
row total column total
table total
Observed counts for gender and TV show:
Gender
Lone Ranger
Boys
Girls
Total
50
50
100
TV Show
Sesame Street
The Simpsons
30
80
110
Total
20
70
90
100
200
300
Find the expected counts for gender and TV show:
Gender
Lone Ranger
TV Show
Sesame Street
The Simpsons
Boys
Girls
The X2 Statistic and its P-value:
Since the expected counts are all large enough to satisfy the required conditions:

How many expected counts are greater than 1? _________

What percent of expected counts are less than 5? _____________
we proceed with the test, comparing the table of observed counts with the table of expected
counts.
We will calculate
O  E
E
2
for each cell in the table and find the sum as our X2 test statistic.
We will still use  2 cdf, but our degrees of freedom are calculated differently.
Degrees of freedom = (# rows - 1)(# columns – 1) = (r - 1)(c - 1)
Enter the observed counts into L1, and the expected counts into L2. Use L3 to calculate the X2
statistic just as we did in the previous section. Think of the X2 statistic as a measure of the
distance that the observed counts are from the expected counts.
Find  2 cdf to determine the probability of gathering data as extreme or more extreme than
this data.
L1
L2
L3
X 2 = ____________
 2 cdf ( _____________________) = _____________
Follow up Analysis:
1. Look at the conditional distribution table (Unusually high? Unusually low?)
2. Look at the chi-squared components (Unusually high?)
3. WARNING: The test confirms only that there is some relationship. The statistical
analysis does not tell us what population our conclusion describes. It cannot generalize to
other TV shows, other ages, etc.
Part 2 - Association/Independence:
The previous example compared 3 TV shows using separate and independent samples. Each
group is a sample from a separate population corresponding to a separate treatment. The null
hypothesis from the Goodness of Fit test of
“______________________________________” took the form of “equal proportions
among the 3 populations.”
The chi square test for association/independence does not compare several populations.
Instead, it classifies observations from a SINGLE population into two or more categories.
EXAMPLE 3:
Many popular businesses are franchises – think of McDonald’s. The owner
of a local franchise benefits from the brand recognition, national advertising, and detailed
guidelines provided by the franchise chain. In return, he or she pays fees to the franchise
firm and agrees to follow its policies. The relationship between the local entrepreneur and the
franchise firm is spelled out in a detailed contract.
One clause that the contract may or may not contain is the entrepreneur’s right to an
exclusive territory. This means that the new outlet will be the only representative of the
franchise in a specified territory and will not have to compete with other outlets of the same
chain. How does the presence of an exclusive-territory clause in the contract relate to
the survival of the business? A study designed to address this question collected data from a
sample of 170 new franchise firms.
Two categorical variables were measured for each firm. First, the firm was classified
as successful or not based on whether or not it was still franchising as of a certain date.
Second, the contract each firm offered to franchises was classified according to whether or
not there was an exclusive-territory clause. Here are the data, arranged in a two-way table:
We are comparing franchises that have exclusive territories with those that do not.
“Exclusive Territory” is the explanatory variable so it’s the column variable. The row variable
is the response variable “success”.
P
H
A
N
T
O
M
S
EXAMPLE 4:
A study of the relationship between men’s marital status and the level of their jobs used data
on all 8235 male managers and professionals employed by a large manufacturing firm. Each
man’s job has a grade set by the company that reflects the value of that particular job to the
company. The authors of the study grouped the many job grades into quarters. Grade 1
contains jobs in the lowest quarter of job grades, and grade 4 contains those in the highest
quarter. Here is the Minitab output for the 4 x 4 table:
Expected counts are printed below observed counts
SINGLE
58
39.08
MARRIED
874
896.44
2
222
173.47
3927
3979.05
70
64.86
20
21.62
4239
3
50
101.90
2396
2337.30
34
38.10
10
12.70
2490
4
7
22.55
533
517.21
7
8.43
4
2.81
551
337
7330
126
42
8235
9.158
13.575
26.432
10.722
+ 0.562
+ 0.681
+ 1.474
+ 0.482
1
Total
ChiSq =
df = 9
Chisquare
DIVORCED WIDOWED
15
8
14.61
4.87
+
+
+
+
0.010
0.407
0.441
0.243
+
+
+
+
2.011
0.121
0.574
0.504
2 cells with expected counts less than 5.0
9.
67.3970
1.0000
Is there a relationship between marital status and job grade?
Total
955
+
+
+
= 67.397
Chapter 14: Inference for Distributions of Categorical Variables
Vocabulary: Chi-square test for Goodness of Fit
Degrees of freedom
Components of Chi-square
Cell counts
Observed count
Cell
Chi-square test for Association/Independence
Chi-square test for Homogeneity of Populations
Chi-square statistic
Expected count
r x c table
14.1 Test for Goodness of Fit
1. What is the chi-square statistic?
2. What is the difference between the notation X2 and
2 ?
3. How many degrees of freedom does a chi-square distribution have?
4. As the chi-square statistic increases, what happens to the P-value?
5. What is the domain of the chi-square distribution?
6. What is the shape of the chi-square distribution?
7. State the null and alternative hypotheses for a Goodness of Fit test.
8. What happens to the shape as the degrees of freedom increase?
14.2 Inference for Two-Way tables
1. What information is contained in a two-way table for a chi-square test?
2. State the null and alternative hypotheses for comparing more than two populations.
3. Explain how to calculate the expected count in any cell of a two-way table.
4. How many degrees of freedom does a chi-square test for a two-way table with “r” rows and
“c” columns have?
5. When is the chi-square test of association/independence used?
6. When is the chi-square test for homogeneity of populations used?
Write a short summary of your performance/effort in the class this year. Include whether or
not you feel prepared for a college-level class.
Inference For Regression (Chp 15)
Learning Targets:
Identify the conditions necessary to do inference for regression
Given a set of data, check that the conditions for doing inference for
regression are present
Explain what is meant by the standard error about the least-squares line
Compute a confidence interval for the slope of the regression line
Conduct a test of the hypothesis that the slope of the regression line is 0
(or that the correlation is 0) in the population
Introduction:
When a scatterplot shows a linear relationship between a quantitative explanatory variable,
______, and a quantitative response variable, ______, we can use the least-squares line fitted
to the data to predict y for a given value of x. Now we want to perform tests and find
confidence intervals in this setting.
Before attempting the inference procedures, we need to examine the data:
1. Make a scatterplot (look for a roughly linear pattern)
2. Find the LSRL (use calculator to find Linear Regression)
3. Look for outliers and influential observations
*outliers are __________________________________________________
*influential observations are ______________________________________
4. Calculate the correlation coefficient and the coefficient of determination (___and ____)
EXAMPLE 1:
One of nature’s patterns connects the percent of adult birds in a colony that
return from the previous year and the number of new adults that join the colony. Here are
data for 13 colonies of sparrow hawks. Examine the data for linear regression
appropriateness.
Percent return (x)
74
66
81
52
73
62
52
45
62
46
60
46
38
New adults (y):
5
6
8
11
12
15
16
17
18
18
19
20
20
Explanatory variable: ___________________ Response variable: ___________________
Is the pattern roughly linear?_______ outliers? ______ influential observations? _______
Equation of LSRL: STAT  CALC  8: ________________________ r = _____ r2 = _____
The Regression Model:
The basic idea here is that there is an “on the average” straight line relationship between y
and x. The true regression line y    x says that the mean response  y moves along a
straight line as x changes.
We can’t observe the true regression line. (  and  are population ______________) The
values of y that we can observe vary about their means according to a __________________.
The standard deviation  determines whether the points fall close to the true regression line
(______  ) or are widely scattered (___________  ).
Calculation of the least squares line  Y  a  bx gives us unbiased estimates for  and  in
the form of ____ and ______.
Conditions for Regression Inference:
We have n observations of an explanatory variable x and a response variable y. Our goal is to
study or predict the behavior of y for given values of x.
Normality 1. For any fixed value of x, the response value y varies according to a _____________.
To verify this, check the residuals’ distribution with a histogram. Is it approximately
normal?
*remember: residual = actual – predicted [RAP]
*need to put predicted values from the LSRL equation into L3.
Then L4 = L2 – L3 (L4 are the residuals)
Independence 2. Repeated observations on the same individual are not allowed.
Linearity 3. The mean response
 y , has a straight-line relationship with x in the form of y    x
where the slope,  , and y-intercept,  , are unknown parameters.
Variability 4. The variability of the responses cannot change with x while the mean response is
changing with x. In other words, the standard deviation of y (we’ll use  ) is the same
for all values of x.
Estimating the Parameters:
We use the LSRL to estimate the parameters
 and  .
The third parameter involved in the
regression model is  . Since the LSRL estimates the true regression line, the residuals from the
LSRL estimate how much y varies about the true regression line. If  is the standard deviation
of responses about the true regression line, an estimate of  is calculated from the residuals in
the sample. A standard deviation calculated from a sample is called the ___________________
s

1
 residual2
n2




1
 y  y 2


n  2 
Why are we using n-2?______________________________________________________
EXAMPLE:
Calculate the standard error about the least squares line for the sparrow hawk
example.
1. LSRL: __________________________________
2.
L1
L2
L3
L4
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
Sum of the squared residuals: ______________
3. s = ____________________
EXAMPLE:
Coffee is a leading export from several developing countries. When coffee
prices are high, farmers often clear forest to plant more coffee trees. Here are five years’
data on prices paid to coffee growers in Indonesia and the percent of forest area lost in a
national park that lies in a coffee-producing region:
Price (cents per pound)
Forest lost (percent)
29
0.49
40
1.59
54
1.69
55
1.82
71
3.10
a) Find the regression equation (LSRL) ________________________________________
b) Explain in words what the slope of the population regression would tell us if we knew it.
c) Use the residuals to estimate the standard deviation of percents of forest lost about the
means given by the population regression line. (Show the formula and the numbers you use)
Confidence Intervals for the Regression Slope:
The slope of the true regression line (population regression) is usually the most important
parameter in a regression problem. The slope is the __________ of increase or decrease of
the mean response as the explanatory variable increases. We often want to estimate the true
slope, so we use ______ of the LSRL as an unbiased estimator. The confidence interval can
show how accurate the estimate is likely to be. To find the confidence interval for the
regression slope, we need the standard error of the slope.
STANDARD ERROR OF THE SLOPE:
SEb
= Estimation of
change in y
change in x
=


1
 y  y 2


n  2 
 (x  x)
2


EXAMPLE:
Does the length of time young children remain at the lunch table help predict
how much they eat? Here are data on 20 toddlers observed over several months at a nursery
school. “Time” is the average number of minutes a child spent at the table when lunch was
served. “Calories” is the average number of calories the child consumed during lunch,
calculated from careful observation of what the child ate each day.
Time:
Calories:
21.4
472
30.8
498
37.7
465
33.5
456
32.8
423
39.5
437
22.8
508
34.1
431
33.9
479
43.8
454
Time:
Calories:
42.4
450
43.1
410
29.2
504
31.3
437
28.6
489
32.9
436
30.6
480
35.1
439
33.0
444
43.7
408
a) Look at the scatterplot of the data and describe briefly what the data show about the
behavior of children
b) Find the equation of the least-squares line for predicting calories consumed from time at
the table.
c) Determine the standard error of the slope.
IN GENERAL:
Confidence interval = estimate  t*(standard error of estimate)
FOR THIS CHAPTER:
FORMULA:
Confidence interval = slope  t*(standard error of slope)
Confidence interval = slope 
t*


1
 y  y 2


n  2 
 (x  x)
2


where t* is the critical value for the (n-2) density curve
EXAMPLE:
Find the confidence interval for the toddler lunch table example above.
Hypothesis Testing:
The most common hypothesis about the slope of  y  x   is:
H0 :   0
This hypothesis says that the slope of the regression line is ___________. If that were
true, we would be saying that the mean of y does not change at all when x changes. In other
words, there is no true linear relationship between x and y. (This does not say “there is no
slope”)
We will follow the PHANTOMS procedure, using a Linear Regression t-test. The test
statistic is just the standardized version of the least-squares slope. It is another t-statistic
of the t(n-2) distribution.
t
slope
b

SE of slope
SEb
EXAMPLE:
Infants who cry easily may be more easily stimulated than others. This may be
a sign of higher IQ. Child development researchers explored the relationship between the
crying intensity of infants four to ten days old and their later IQ test scores.
Infants’ crying and IQ scores
L1
crying
L2
IQ
L1
crying
L2
IQ
L1
crying
L2
IQ
10
12
9
16
18
15
12
20
16
33
19
18
22
87
97
103
106
109
114
119
132
136
159
103
112
135
20
16
23
27
15
21
12
15
17
13
13
16
30
90
100
103
108
112
114
120
133
141
162
104
118
155
12
12
14
10
23
9
16
31
22
17
18
19
94
103
106
109
113
119
124
135
157
94
109
120
Perform a significance test to determine if a linear relationship exists between the crying
intensity and IQ scores
P
H
A
(a) Independent observations?
(b) Linear relationship? (Look at the scatterplot)
(c) Normality?
(d) Variation consistant?
N
T
O
M
S
When reading a computer output of regression information, be aware that the software
usually gives a 2-sided p-value. If we have a one-tailed test, we will need to _____________.
Chapter 15: Inference for Regression
Vocabulary: Explanatory variable
True regression line
Standard error about the line
Response variable
Parameters , , 
Residuals
1. Explain the 4 conditions for regression inference.
2. Explain the difference between the equations
y   x and y  a  bx
3. Explain the concept of standard error about the least-squares line
4. What does the slope  represent?
5. How does the null hypothesis H0 :   0 fit in with the other null hypotheses we have used?
6. What does the confidence interval estimate in this chapter?