Please complete the Prerequisite skills on PG
742 #1-8
Big ideas:

Finding measures of central tendency and dispersion

Using normal distributions

Working with samples
Essential Question:
When will the mean of a data
set differ significantly from the
median of the data set?


Statistics: Numerical values used to
summarize and compare sets of data
Standard deviation: The typical
difference between a data value and the
mean.
EXAMPLE 1
Find measures of central tendency
Waiting Times
The data sets at the right give the waiting times (in minutes) of several
people at two veterinary offices. Find the mean, median, and mode of
each data set.
SOLUTION
Median: 20 Mode: 24
=
14 + 17 +
+ 32
9
…
Office A: Mean: x
=
198
22
9 =
EXAMPLE 1
Find measures of central tendency
Office B: Mean: x
Median: 18 Mode: 18
8 + 11 + …
+ 23
=
9
144
16
=
9 =
for Example 1
GUIDED PRACTICE
TRANSPORTATION
1.
The data set below gives the waiting times (in minutes) of 10
students waiting for a bus. Find the mean, median, and mode of the
data set.
4, 8, 12, 15, 3, 2, 6, 9, 8, 7
SOLUTION
Mean: x
…
4 + 8 + 12 + + 7
=
10
Median: 7.5 Mode: 8
=
74
10
= 7.4
EXAMPLE 2
Find ranges of data sets
Find the range of the waiting times in each data set.
SOLUTION
Office A: Range = 32 – 14
Office B: Range = 23 – 8
= 18
= 15
Because the range for office A is greater, its waiting times are more
spread out.
EXAMPLE 3
Standardized Test Practice
SOLUTION
Office A:
Office B:
ANSWER
=
=
(14 – 22)2 + (17 – 22)2 + ... + (32 – 22)2
9
(8 –16)2 + (11 16)2– +
9
The correct answer is D.
...
+ (23 16)2
290
=
9
182
–
=
9
5.7
4.5
for Examples 2 and 3
GUIDED PRACTICE
2.
Find the range and standard deviation of
set.
4, 8, 12, 15, 3, 2, 6, 9, 8, 7
SOLUTION
Range = 15 – 2
Standard deviation
= 13
= 3.8
the data
When will the mean of a data set
differ significantly from the median
of the data set?
A data set with an outlier will
show a greater difference
between the mean and the
median than will the data set
with the outlier excluded

What happens to the average when an outlier
(much higher than the other values) is part of
the data set?
Essential Question:
Are all statistics of a data set
affected when you transform
the values in a data set?




Mean: The “average” of a set of numbers
Median: The middle number of a set of
numbers
Mode: The number that occurs most often
in a set of number
Range: A measure of dispersion equal to
the difference between the greatest and
least data values

Adding a Constant to Data Values:
◦ The mean, median, and mode of the new data set
can be obtained by adding the same constant to the
mean, median and mode of the original data set.
◦ The range and standard deviation are unchanged.

Multiplying Data Values by a Constant:
◦ When each value of a data set is multiplied by a
constant, the new mean, median, mode, range, and
standard deviation can be found by multiplying
each original statistic by the same constant.
Are all statistics of a data set
affected when you transform the
values in a data set?
The range and standard
deviation are not affected by an
addition transformation.
EXAMPLE 1
Add a constant to data values
Astronauts
The data set below gives the weights (in pounds) on Earth of eight
astronauts without their space suits. A space suit weighs 250 pounds
on Earth.Find the mean, median, mode, range, and standard deviation
of the weights of the astronauts without their space suits and with their
space suits.
142, 150, 155, 156, 160, 160, 166, 175
EXAMPLE 1
SOLUTION
Add a constant to data values
EXAMPLE 2
Multiply data values by a constant
OLYMPICS
The data set below gives the winning distances (in meters) in the men’s
Olympic triple jump events from 1964 to 2004. Find the mean,
median,mode, range, and standard deviation of the distances in meters
and of the distances in feet. (Note: 1 meter 3.28 feet.)
16.85, 17.39, 17.35, 17.29, 17.35, 17.26, 17.61, 18.17,
18.09, 17.71, 17.79
EXAMPLE 2
SOLUTION
Multiply data values by a constant
GUIDED PRACTICE
1.
for Examples 1 and 2
Astronauts: The Manned Maneuvering Unit (MMU) is equipment that
latches onto an astronaut’s space suit and enables the astronaut to
move around outside the spacecraft. The MMU weighs about 300
pounds on Earth. Find the mean, median, mode, range, and standard
deviation of the weights of the astronauts in Example 1 with their
space suits and MMUs.
142, 150, 155, 156, 160, 160, 166, 175
for Examples 1 and 2
GUIDED PRACTICE
SOLUTION
Mean
Median
708
708
Mode
710
Range
33
Standard
deviation
9.3
GUIDED PRACTICE
for Examples 1 and 2
2. What If? In Example 2, find the mean, median, mode, range, and
standard deviation of the distances in yards. (Note: 1 meter 1.09
yards.)
SOLUTION
Mean
Median
Mode
Range
Standard
deviation
19.11
18.96
18.91
1.44
0.40
Are all statistics of a data set
affected when you transform the
values in a data set?
The range and standard
deviation are not affected by an
addition transformation.

What changes when you multiply by a
constant?
Essential Question:
Where are the values in a
normal distribution that rarely
occur displayed on a normal
curve?


Normal distribution: A probability distribution with
mean, x and standard deviation modeled by a bellshaped curve
Normal curve: A smooth, symmetrical, bell-shaped
curve that can model normal distributions and
approximate some binomial distributions

Standard normal distribution: The normal
distribution with mean 0 and standard deviation 1

Z-score: The number z of standard deviations that
a data value lies above or below
EXAMPLE 1
Find a normal probability
A normal distribution has mean x and standard deviation σ. For a
randomly selected x-value from the distribution, find P(x – 2σ ≤ x ≤ x).
SOLUTION
The probability that a randomly selected x-value
lies between – 2σ and is the shadedxarea
under the normal curve shown.
x
P( x – 2σ ≤ x ≤ x) = 0.135 + 0.34 = 0.475
EXAMPLE 2
Interpret normally distribute data
Health
The blood cholesterol readings for a group of women are normally
distributed with a mean of 172 mg/dl and a standard deviation of 14
mg/dl.
a.
b.
About what percent of the women have readings between 158 and
186?
Readings higher than 200 are considered undesirable. About what
percent of the readings are undesirable?
EXAMPLE 2
Interpret normally distribute data
SOLUTION
a. The readings of 158 and 186 represent one
standard deviation
on either side of the mean,
as shown below. So, 68% of the
women have
readings between 158 and 186.
EXAMPLE 2
Interpret normally distribute data
b. A reading of 200 is two standard deviations to the
right of the
mean, as shown. So, the percent of
readings that are undesirable
is 2.35% + 0.15%, or
2.5%.
for Examples 1 and 2
GUIDED PRACTICE
x deviation σ. Find the
A normal distribution has mean and standard
indicated probability for a randomly selected x-value from the
distribution.
1. P( x ≤ x )
ANSWER
0.5
for Examples 1 and 2
GUIDED PRACTICE
2.
P( x > x )
ANSWER
0.5
for Examples 1 and 2
GUIDED PRACTICE
3.
P( x <
ANSWER
x<
x+ 2σ )
0.475
for Examples 1 and 2
GUIDED PRACTICE
4.
P( x– σ < x <
ANSWER
x)
0.34
for Examples 1 and 2
GUIDED PRACTICE
5. P(x ≤ x – 3σ)
ANSWER
0.0015
for Examples 1 and 2
GUIDED PRACTICE
6. P(x > x + σ)
ANSWER
0.16
for Examples 1 and 2
GUIDED PRACTICE
7.
WHAT IF? In Example 2, what percent of the women have
readings between 172 and 200?
ANSWER
47.5%
EXAMPLE 3
Use a z-score and the standard normal table
Biology
Scientists conducted aerial surveys of a seal sanctuary and recorded the
number x of seals they observed during each survey. The numbers of
seals observed were normally distributed with a mean of 73 seals and a
standard deviation of 14.1 seals. Find the probability that at most 50
seals were observed during a survey.
EXAMPLE 3
Use a z-score and the standard normal table
SOLUTION
STEP 1
Find: the z-score corresponding to an x-value of 50.
50 – 73
z= x – x =
14.1
STEP 2
Use: the table to find P(x < 50)
–1.6
P(z < – 1.6).
The table shows that P(z < – 1.6) = 0.0548. So, the
probability that at most 50 seals were observed during a
survey is about 0.0548.
EXAMPLE 3
Use a z-score and the standard normal table
GUIDED PRACTICE
8.
for Example 3
WHAT IF? In Example 3, find the probability that at most 90 seals
were observed during a survey.
ANSWER
0.8849
GUIDED PRACTICE
9.
for Example 3
REASONING: Explain why it makes sense that
P(z < 0) = 0.5.
ANSWER
A z-score of 0 indicates that the z-score and the mean are the
same. Therefore, the area under the normal curve is divided into
two equal parts with the mean and the z-score being equal to
0.5.
Where are the values in a normal
distribution that rarely occur
displayed on a normal curve?
The values occur at the end of
the curve

The mean score on an exam was 78. You scored
within 5 points of the mean. If x = 78 +/- 5
represents your possible score x on the exam, what
is the range of your score?
Essential Question:
What should be true of the
sample when you conduct a
survey?

No new vocabulary for this lesson.
EXAMPLE 1
Classify samples
Baseball
A sportswriter wants to survey college baseball coaches about whether
they think wooden bats should be mandatory throughout college baseball.
Identify the type of sample described.
a.
The sportswriter contacts only the coaches that he has cell phone
numbers for in order to get quick responses.
b.
The sportswriter mails out surveys to all the
coaches and uses only the surveys that are
returned.
EXAMPLE 1
Classify samples
SOLUTION
a.
The sportswriter selected coaches that are easily accessible. So, the
sample is a convenience sample.
b.
The coaches can choose whether or not to respond. So, the sample is
a self-selected sample.
EXAMPLE 2
Identify a biased sample
Concert Attendance
The manager of a concert hall wants to know how often people in the
community attend concerts. The manager asks 50 people standing in line for
a rock concert how many concerts per year they attend. Tell whether the
sample is biased or unbiased. Explain your reasoning.
SOLUTION
The sample is biased because people standing in line for a rock
concert are more likely to attend concerts than people in general.
EXAMPLE 3
Choose an unbiased sample
Senior Class Prom
You are a member of the prom committee. You want to poll members of the
senior class to find out where they want to hold the prom. There are 324
students in the senior class. Describe a method for selecting a random
sample of 40 seniors to poll.
SOLUTION
STEP 1
Make: a list of all 324 seniors. Assign each senior a different
integer from 1 to 324.
EXAMPLE 3
STEP 2
Choose an unbiased sample
Generate: 40 unique random integers from
1 to 324 using the randInt feature of a graphing
calculator. The screen at the
right shows six such random integers.
If while generating the integers you obtain
a duplicate, discard it and generate a new, unique integer
as a replacement.
STEP 3
Choose: the 40 students that correspond to the 40
integers you generated in Step 2.
GUIDED PRACTICE
1.
for Examples 1, 2, and 3
SCHOOL WEBSITE: A computer science teacher wants to know if
students would like the morning announcements posted on the
school’s website. He surveys students in one of his computer
science classes. Identify the type of sample described, and tell
whether the sample is biased.
SOLUTION
The computer science teacher surveys students in one of his
computer science classes where students are easy to reach. So, the
sample is a convenience sample. The sample is biased as the
students of only the computer class are surveyed.
GUIDED PRACTICE
for Examples 1, 2, and 3
2. WHAT IF? In Example 3, what is another method you could use to
generate a random sample of 40 students?
SOLUTION
Place each student’s name on a small piece of paper in a hat and
draw 40 names.
What should be true of the sample
when you conduct a survey?
The survey should include a
random sample of the
population of interest and the
sample should be large enough
to guarantee a reasonable
margin of error.