Statistical Reasoning
Preview Assignment 12.C
Preview Assignment 12.C
Preparing for the next class
In the next lesson, you will need to find normal distribution probabilities using a table of
areas under the standard normal curve and interpret the meaning of z-scores.
Find the two-page table of probabilities at the end of this preview and have it ready to
use for this assignment. Also be sure to bring the table to class for your next lesson.
Suppose a variable has a normal distribution. To find the probability of an observation
having a z-score of less than 1.23, think about the normal distribution as follows:
Write 1.23 as 1.2 + 0.03 and locate 1.2 on the left side of the table as shown.
The Charles A. Dana Center at
The University of Texas at Austin
1
Version 2.0
Statistical Reasoning
Preview Assignment 12.C
Next, locate 0.03 at the top of the table and read the table as illustrated below.
There is a 0.891 probability of having a z-score less than 1.23, which can be written as
P(z < 1.23) = 0.891.
Note that a positive z-score is to the right of the mean, and the probability is more than
0.5. (Look at the shaded curve at the beginning of this assignment.)
1)
Use the table above to determine the probability of z < 1.14.
2)
Use Table 1 at the end of this preview to determine the probability of z < .36.
3)
Use Table 2 at the end of this preview to determine the probability of z < –1.63.
The Charles A. Dana Center at
The University of Texas at Austin
2
Version 2.0
Statistical Reasoning
Preview Assignment 12.C
Using Table 1 or Table 2 at the end of this preview, determine the following
probabilities.
4)
P(z < 1.19) =
5)
P(z < –1.19) =
6)
P(z < 3.02) =
7)
P(z < –3.02) =
8)
Without looking at the table, do you expect the probability of z < –0.4 to be:
a)
Less than 0
b)
Between 0 and 0.5
c)
Between 0.5 and 1
d)
More than 1
The Charles A. Dana Center at
The University of Texas at Austin
3
Version 2.0
Statistical Reasoning
Preview Assignment 12.C
Monitoring your readiness
9)
To effectively plan and use your time wisely, it helps to think about what you know
and do not know. For each of the following, rate how confident you are that you
can successfully do that skill. Use the following descriptions to rate yourself:
5—I am extremely confident I can do this task.
4—I am somewhat confident I can do this task.
3—I am not sure how confident I am.
2—I am not very confident I can do this task.
1—I am definitely not confident I can do this task.
Skills needed for Lesson 12, Part C
Skill or Concept: I can . . .
Questions to check your
understanding
Find normal distribution
probabilities using tables.
Rating
from 1 to 5
1–7
Interpret the meaning of
z-scores.
8
Now use the ratings to get ready for your next lesson. If your rating is a 3 or below,
you should get help with the material before class. Remember, your instructor is
going to assume that you are confident with the material and will not take class time to
answer questions about it.
10) What are some ways to get help?
The Charles A. Dana Center at
The University of Texas at Austin
4
Version 2.0
Statistical Reasoning
Preview Assignment 12.C
Table 1:
The Charles A. Dana Center at
The University of Texas at Austin
5
Version 2.0
Statistical Reasoning
Preview Assignment 12.C
Table 2:
The Charles A. Dana Center at
The University of Texas at Austin
6
Version 2.0
Statistical Reasoning
Student Pages 12.C, Probability and critical values
Lesson 12, Part C
Probability and Critical Values
You are part of a design team working for
an American auto company that is
attempting to make the most fuel-efficient
car on the market.
1)
What are some features of a car that
might affect fuel efficiency?
Credit: ©iStock.com
Objectives for the lesson
You will understand:
o The z-score as it relates to the standard normal curve.
You will be able to:
o Calculate probabilities using the normal curve.
In designing the car, you will need to be sure that most people will be comfortable.
Begin by analyzing the heights of adult American females. (You will analyze male
heights in the homework). Women’s heights are normally distributed with a mean of 64
inches and a standard deviation of 2.8 inches.
2)
Because of the design of the interior, women who are less than 60 inches tall will
not be comfortable in the car.
Part A: Draw a sketch of the normal distribution of female heights. Shade the area
under the curve that represents the probability that a woman is less than 60 inches
tall.
Part B: Determine the probability that a randomly selected adult American female
is less than 60 inches tall.
The Charles A. Dana Center at
The University of Texas at Austin
7
Version 2.0
Statistical Reasoning
Student Pages 12.C, Probability and critical values
3)
Women who are over 70 inches tall will also be uncomfortable in the car.
Part A: Shade the area under the curve that corresponds to adult American
females who are less than 70 inches tall.
Part B: Determine the probability that a randomly selected adult American female
is less than 70 inches tall.
4)
Now focus on the women who will be comfortable.
Part A: Shade the area under the curve that corresponds to adult American
females who are between 60 inches tall and 70 inches tall.
Part B: Determine the probability that a randomly selected adult American female
is between 60 inches tall and 70 inches tall.
5)
If the prototype comfortably accommodates adult females between 60 inches tall
and 70 inches tall, what proportion of adult American women will be comfortable in
the car?
6)
What proportion of adult American females are too tall to fit in the prototype
comfortably?
The Charles A. Dana Center at
The University of Texas at Austin
8
Version 2.0
Statistical Reasoning
Suggested Instructor Notes 12.C, Probability and critical values
Lesson 12, Part C
Probability and Critical Values
Overview and student objectives
Lesson Length: 25 minutes
Overview
Prior Lesson: Lesson 12,
Part B, “z-Scores and
Normal Distributions”
In this lesson, students will analyze normally distributed
heights of women in order to make automobile design
recommendations. In Preview Assignment 12.C, students
computed probabilities using z-tables; they will apply that skill
in this lesson to compute the probabilities of women meeting
certain height requirements using the normal curve. The
practice assignment repeats these concepts for men.
Next Lesson: Lesson 12,
Part D, “Probability and
Critical Values (continued)”
(25 minutes)
Objectives
Outcomes: PS2, PS3
Students will understand:
Goals: Communication and
Problem Solving
•
The z-score as it relates to the standard normal curve.
Students will be able to:
•
Constructive Perseverance
Level: 2
Related Foundations
Outcome: N8
Calculate probabilities using the normal curve.
Suggested resources and preparation
Materials and technology
•
Computer, projector, document camera
•
Preview Assignment 12.C
•
Student Pages for Lesson 12, Part C
•
Practice Assignment 12.C
Prerequisite assumptions
Before beginning this lesson, students should be able to find probabilities using
z-tables.
Making connections
This lesson:
•
Connects back to computing z-scores.
•
Connects forward to sampling distributions and the Central Limit Theorem.
The Charles A. Dana Center at
The University of Texas at Austin
9
Version 2.0
Statistical Reasoning
Suggested Instructor Notes 12.C, Probability and critical values
Background context
The students used a standard normal table in the preview and were asked to bring it to
class.
Suggested instructional plan
Frame the lesson
(6 minutes)
Student
Pages
Literacy
Support
•
Debrief the preview assignment to ensure student understanding. Ask
students to refer to the z-table provided in the preview. (They were
asked to bring the table to class.)
•
Choose a positive z-score and a negative z-score example to write on
the board. Ask students to determine the probability of a z-score less
than those given.
•
Look for students who do not understand how to use a table to find
probabilities and provide assistance before moving on.
•
Take responses to question 1.
•
“How do car companies make decisions about the interior design of
an automobile?”
•
“What is a ‘prototype’?”
Have students share what prototype means and reach consensus on
the definition before moving on.
•
“Why do you think it’s important to design an automobile that meets
the needs of most people?”
•
Allow discussion before transitioning to the lesson.
•
Transition to the lesson activities by briefly discussing the Objectives
for the lesson.
Lesson activities
(15 minutes)
Questions 2–3
Guiding
Question
•
Note: Consider having half of the class complete problem 2 and the
other half of the class complete problem 3. Use the results from each
group to answer question 4.
•
“How could we label the x-axis to help us visualize standard
deviations?”
•
Assist students with drawing the normal distribution. Ensure that
students are labeling appropriately.
The Charles A. Dana Center at
The University of Texas at Austin
10
Version 2.0
Statistical Reasoning
Suggested Instructor Notes 12.C, Probability and critical values
•
Assist students, as needed, in locating and plotting 60 on their graph.
•
“Where is 60 on the x-axis?”
•
“What information do you need in order to determine the shaded
area?” [Answer: The z-score for 60.]
•
Facilitate question 3 in a similar manner.
Question 4
•
This question may be a bit more difficult for students because it
involves visualizing two different areas and subtracting.
•
Have the two groups share the results and compare the areas.
o “How can I use both areas to find the area in between?”
o “In terms of area, what information does the table provide?”
o “How can I use both z-scores to help me find the answer?”
•
Remind students that the table only provides the area to the left of a
z-score and guide them to understand that subtraction is the key. You
may want display a student’s graph, point to the area in question, and
guide students to understand how to subtract areas in order to arrive
at the answer.
•
If students used shading and double-shading on a single graph,
consider drawing separate curves on the board, with the graph for
question 3 aligned beneath the curve for question 2. Label the areas
on each curve. Draw a third graph underneath with vertical lines at 60
and 70, then shade in between the lines.
Question 5
•
This question promotes discussion of the relationship between
proportions and probabilities. For example, the probability that a
woman is less than 60 inches tall follows directly from the proportion
of the population of women who are less than 60 inches tall.
Question 6
•
This question requires students to subtract from 1 to find the area
above a z-score. Ask:
o “What is the total area under the curve?” [Answer: 1]
o “If I know the area below, how can I find the area above?”
•
If students are struggling, refer them back to the curve and shading
for question 3. Again, if the students used shading and doubleshading, refer them to the individual graphs on the board.
Guiding
Questions
The Charles A. Dana Center at
The University of Texas at Austin
11
Version 2.0
Statistical Reasoning
Suggested Instructor Notes 12.C, Probability and critical values
Wrap-up/transition
(4 minutes)
Wrap-up
Transition
•
Remind students that the normal distribution occurs much more often
than uniform distributions. Explain that the normal distribution will also
play a role as they move into the idea of inference and sampling.
•
Remind students that this study focused only on women. Men will be
analyzed in their homework.
•
Have students refer back to the Objectives for the lesson and check
the ones they recognize from the activity. Alternatively, they may
check the objectives throughout the lesson.
•
“Management sends a memo: The car must be comfortable for all but
the tallest 2.5% of American women. What should we do?”
Suggested assessment, assignments, and reflections
•
Give Practice Assignment 12.C.
•
Give the Preview Assignments, if any, for the lesson activities you plan to
complete in the next class meeting.
The Charles A. Dana Center at
The University of Texas at Austin
12
Version 2.0
Statistical Reasoning
Practice Assignment 12.C
Practice Assignment 12.C
1)
The heights of adult American males are approximately normally distributed with a
mean of 69 inches and a standard deviation of 2.5 inches.
Part A: What is the probability that a randomly selected adult American male is
less than 63 inches tall? Round your answer to the nearest thousandth.
Part B: What is the probability that a randomly selected adult American male is
less than 76 inches tall? Round your answer to the nearest thousandth.
Part C: What is the probability that a randomly selected adult American male is
more than 76 inches tall? Round your answer to the nearest thousandth.
Part D: What is the probability that a randomly selected adult American male is
between 63 and 76 inches tall? Round your answer to the nearest thousandth.
13
Version 2.0
Statistical Reasoning
Practice Assignment 12.C
2)
Is there a difference between the following two questions? Explain.
What is the probability that a randomly selected adult American male is more than
76 inches tall?
What proportion of adult American males are more than 76 inches tall?
3)
The time students take to complete a statistics test is approximately normally
distributed with a mean of 74 minutes and a standard deviation of 8 minutes.
Part A: What is the probability that a student took more than 1 hour to complete the
test?
Part B: What is the probability that a student took less than 84 minutes to complete
the test?
Part C: What is the probability that a student took between 60 and 84 minutes to
complete the test?
14
Version 2.0
Statistical Reasoning
Practice Assignment 12.C
4)
A company regularly chooses 100 people to survey through random selection.
They want to ensure that their survey groups are reasonably representative of the
U.S. population. 17% of the U.S. population is Hispanic. If the company truly used
simple random sampling of the U.S. population, the number of Hispanics in the
survey groups would follow the probability distribution shown below.
These data look roughly normal, so we can approximate this with a normal
distribution with the same mean and standard deviation.
Assuming the number of Hispanics in the survey groups followed this continuous
distribution, answer following the questions.
Part A: What percentage of the survey groups have fewer than 20 Hispanics?
15
Version 2.0
Statistical Reasoning
Practice Assignment 12.C
Part B: What percentage of survey groups have more than 18 Hispanics?
Part C: What percentage of survey groups have between 15 and 25 Hispanics?
Part D: What percentage of survey groups have fewer than 6 Hispanics? What
might you suspect if most of the company's survey groups actually had fewer than
6 Hispanics?
16
Version 2.0
Statistical Reasoning
Practice Assignment 12.C
5)
For each of the following situations:
o Sketch a normal curve and shade a region that represents the probability.
o Calculate the probability.
The first problem has been done as an example.
Example: If µ = 25 and σ = 3, the probability that x > 20.
Answer: P(x > 20) = 0.9522
Part A: If µ = 413 and σ = 63, the probability that x < 400.
17
Version 2.0
Statistical Reasoning
Practice Assignment 12.C
Part B: If µ = –40 and σ = 5, the probability that x > –37.
Part C: If µ = 0.16 and σ = 0.0132, the probability that 0.16 < x < 0.17.
18
Version 2.0
Statistical Reasoning
Preview Assignment 12.D
Preview Assignment 12.D
Preparing for the next class
In the next lesson, you will need to use a normal probability table to find the z-score
associated with a certain probability, read and understand a probability table, and solve
equations.
In the last lesson, you used z-scores to find a probability. Now you will read the table
“from the inside out,” starting from a probability and finding the z-score.
To determine the z-score associated with probability = 0.936, begin by finding 0.936 in
the body of the table. Then look left and up to determine the z-score associated with the
probability. See below:
Probability = 0.936 is associated with the z-score 1.52.
The Charles A. Dana Center at
The University of Texas at Austin
19
Version 2.0
Statistical Reasoning
Preview Assignment 12.D
1)
Find the z-score that is associated with a probability of 0.841 in the table.
2)
Which of the statements is true?
a)
The probability that z > 1.00 is 0.841.
b)
The probability that z < 1.00 is 0.841.
c)
The probability z = 1.00 is 0.841.
3)
Find the z-score that is associated with a probability = 0.053 in Table 2 at the end
of this preview.
4)
Which of the following is true:
a)
The probability that z > –1.62 is 0.053.
b)
The probability that z < –1.62 is 0.053.
c)
The probability z = –1.62 is 0.053.
Solve the following equations:
5)
Solve –3.4 = x – 8.
The Charles A. Dana Center at
The University of Texas at Austin
20
Version 2.0
Statistical Reasoning
Preview Assignment 12.D
x
.
7
6)
Solve 2.5 =
7)
Solve 5 =
8)
Solve 33.4 =
x−2
.
3
x−23.2
.
! 5
Monitoring your readiness
9)
To effectively plan and use your time wisely, it helps to think about what you know
and do not know. For each of the following, rate how confident you are that you
can successfully do that skill. Use the following descriptions to rate yourself:
5—I am extremely confident I can do this task.
4—I am somewhat confident I can do this task.
3—I am not sure how confident I am.
2—I am not very confident I can do this task.
1—I am definitely not confident I can do this task.
The Charles A. Dana Center at
The University of Texas at Austin
21
Version 2.0
Statistical Reasoning
Preview Assignment 12.D
Skills needed for Lesson 12, Part D
Skill or Concept: I can . . .
Questions to check your
understanding
Use a normal probability table
to find the z-score associated
with a certain probability.
1, 3
Read and understand a
probability table.
2, 4
Solve equations.
5–8
Rating
from 1 to 5
Now use the ratings to get ready for your next lesson. If your rating is a 3 or below,
you should get help with the material before class. Remember, your instructor is
going to assume that you are confident with the material and will not take class time to
answer questions about it.
10) What are some ways to get help?
The Charles A. Dana Center at
The University of Texas at Austin
22
Version 2.0
Statistical Reasoning
Preview Assignment 12.D
Table 1:
The Charles A. Dana Center at
The University of Texas at Austin
23
Version 2.0
Statistical Reasoning
Preview Assignment 12.D
Table 2:
The Charles A. Dana Center at
The University of Texas at Austin
24
Version 2.0
Statistical Reasoning
Student Pages 12.D, Probability and critical values (continued)
Lesson 12, Part D
1)
Probability and Critical Values (continued)
The manufacturer wants the design
team to make the car comfortable for
all but the tallest 2.5% of American
women. Is it possible to determine the
height limit that would meet this
requirement? Explain.
Credit: @iStock.com
Objectives for the lesson
You will understand:
o The z-score as it relates to the standard normal curve.
You will be able to:
o Calculate probabilities using the normal curve.
o Find critical values using the normal curve.
Recall that the heights of American women are approximately normally distributed with
a mean of 64 inches and a standard deviation of 2.8 inches.
2)
How can you use a normal curve to represent the tallest 2.5% of adult American
females? How would you use this information in the probability table?
3)
Recall that z =
4)
Suppose the design team also decides to make the car comfortable for all but the
shortest 2.5% of females. Sketch the graph of the normal distribution and shade
the area needed to find the z-score.
5)
What is the cutoff height for the shortest 2.5% of women?
x−µ
. What is the cutoff height for the tallest 2.5% of women?
σ
The Charles A. Dana Center at
The University of Texas at Austin
25
Version 2.0
Statistical Reasoning
Student Pages 12.D, Probability and critical values (continued)
6)
If the car is comfortable for all women who have heights between your answers
from questions 3 and 5, what proportion of American women fit comfortably in the
car?
The Charles A. Dana Center at
The University of Texas at Austin
26
Version 2.0
Statistical Reasoning
Suggested Instructor Notes 12.D, Probability and critical values (continued)
Lesson 12, Part D
Probability and Critical Values (continued)
Overview and student objectives
Lesson Length: 25 minutes
Overview
In this lesson, students analyze normally distributed
heights of American women. Constraints are introduced
that require that the car being designed accommodate only
the middle 95% of consumers. Students work backward
from a given probability to find the upper and the lower
cutoff heights for women.
Students may need additional support in this lesson. In
particular, working backward with the z-score formula may
require some direct instruction.
Objectives
Next Lesson: Lesson 13, Part
A, “Sampling Variability” (25
minutes)
Constructive Perseverance
Level: 3
Outcomes: PS2, PS3
Goals: Communication and
Problem Solving
Related Foundations
Outcomes: N8, A3
Students will understand:
•
Prior Lesson: Lesson 12, Part
C, “Probability and Critical
Values”
The z-score as it relates to the standard normal curve.
Students will be able to:
•
Calculate probabilities using the normal curve.
•
Find critical values using the normal curve.
Suggested resources and preparation
Materials and technology
•
Computer, projector, document camera
•
Preview Assignment 12.D
•
Student Pages for Lesson 12, Part D
•
Practice Assignment 12.D
•
Students should have their z-table from Preview Assignment 12.C. You may wish
to have a few extra copies available.
Prerequisite assumptions
Before beginning this lesson, students should be able to:
•
Find probabilities using z-tables.
•
Manipulate algebraic equations to solve for unknowns.
The Charles A. Dana Center at
The University of Texas at Austin
27
Version 2.0
Statistical Reasoning
Suggested Instructor Notes 12.D, Probability and critical values (continued)
Making connections
This lesson:
•
Connects back to computing z-scores.
•
Connects forward to sampling distributions and the Central Limit Theorem.
Background context
None.
Suggested instructional plan
Frame the lesson
(6 minutes)
Question 1
Motivating
Questions
Literacy
Support
•
“Is it possible to build a car that accommodates all people?”
•
“Is it realistic for car companies to build cars that will accommodate
all people, or do they target certain people?”
•
“What are some reasons that it’s difficult for car companies to build a
car that accommodates all people?”
•
Encourage class discussion. Explain to students that car companies
may place constraints on car designs because of expectations of fuel
efficiency, cost of materials, and assembly line limitations.
•
“What is a ‘constraint’?”
•
Have students discuss the definition and reach a consensus on the
definition before moving on. Connect constraints to the upper and
lower cutoff scores that students will be finding in this lesson.
•
Transition to the lesson activities by briefly discussing the
Objectives for the lesson.
Lesson activities
(15 minutes)
Question 2
Group
Work
•
Encourage students often to turn to a quick, shaded, normal curve
when they are unsure how to begin a problem.
•
Stress, whenever possible, the connection between the percentage
of a population, the proportion of a population, and the probability of
selecting a particular value from a population data set.
The Charles A. Dana Center at
The University of Texas at Austin
28
Version 2.0
Statistical Reasoning
Suggested Instructor Notes 12.D, Probability and critical values (continued)
Guiding
Questions
•
“What does this normal curve represent?” [Answer: The population of
adult American women.]
•
“Where in the curve would the tallest 2.5% of women be?” [Answer:
In the right tail.]
•
“Estimate where the boundary would be and mark with a z. Shade
that area of the curve. What is the probability that a randomly
selected American woman has a z-score larger than that?” [Answer:
0.025.]
One valid strategy would be to look up 0.025 in the table, get
z = –1.96, and use the symmetry of the table to state that the z-score
should be +1.96.
•
Alternatively, facilitate the following strategy:
o “What is the probability that a randomly selected American
woman has a z-score smaller than this z?” [Answer: 1 – 0.025 or
0.975.]
o “What is that z-score?” [Answer from the table: z = 1.96.]
Question 3
•
Students may struggle a bit more because this question involves
using z-score formula to solve for the height. The preview
assignment contained two basic equation-solving tasks.
•
Encourage students to always write down the symbolic formula
x−µ
before substituting in any values. Encourage them to think
z=
σ
through the connection in words as well.
•
“If we knew a specific height, where does it go in the formula?”
[Answer: In place of x.]
•
“In this case. we are looking for a cutoff height. Which variable
represents height?” [Answer: The x-variable represents height.]
•
“Of the variables in the equation, which are known and which are
unknown?” [Answer: We know the mean and the standard deviation.
We also have the target proportion so we can use the table to
determine the z-score.]
•
“What do you need to do in order to isolate x?” [Answer: Multiply
both sides of the equation by the standard deviation, then add the
mean to both sides.]
Question 4
•
Students should be able to work independently on this question.
•
You might want to pause the lesson at this point and have students
describe the step-by-step process they just went through. Have a
student write the list on the board for students to refer to.
The Charles A. Dana Center at
The University of Texas at Austin
29
Version 2.0
Statistical Reasoning
Suggested Instructor Notes 12.D, Probability and critical values (continued)
•
For example, the list on the board could look like this:
o Step 1: Shade the appropriate area. (The table values refer to the
area to the left of the associated z-score.)
o Step 2: Find the probability in the body of the table and determine
the corresponding z-score.
o Step 3: Use the formula to solve for x.
Question 5
•
Students have already calculated the z-score to be –1.96 for 2.5%;
x−µ
to calculate the
therefore, they just need to use the formula z =
σ
cutoff for the shortest height.
Question 6
•
This question synthesizes the results of the activity. If students
struggle, suggest they draw a new curve and shade the area that
represents the heights of the women who will fit in the car (the
middle portion of the curve, rather than the two tails, as was done in
the activity).
Wrap-up/transition
(4 minutes)
Wrap-up
•
Close by asking students the following questions:
o “Would a woman who is 72 inches tall fit comfortably in the car?”
[Answer: No, the cutoff height was 69.49 inches, so she would be
a little bit too tall. Her head may hit the roof.]
o “Would a woman who is 56 inches tall fit comfortably in the car?”
[Answer: No, the cutoff height was 58.51 inches, so she would be
a little bit too short. Her feet might not reach the pedals.]
o “Would you fit comfortably in the car? Why or why not?”
•
Take some time to compare and contrast the uniform distributions
studied earlier to the normal distribution. Remind students about the
importance of the normal distribution. Many things in real life have a
normal distribution.
•
Highlight the idea of a business, in this case a car manufacturer,
using statistics to make business decisions.
•
Have students refer back to the Objectives for the lesson and
check the ones they recognize from the activity. Alternatively, they
may check the objectives throughout the lesson.
The Charles A. Dana Center at
The University of Texas at Austin
30
Version 2.0
Statistical Reasoning
Suggested Instructor Notes 12.D, Probability and critical values (continued)
Suggested assessment, assignments, and reflections
•
Give Practice Assignment 12.D.
•
Give the Preview Assignments, if any, for the lesson activities you plan to
complete in the next class meeting.
The Charles A. Dana Center at
The University of Texas at Austin
31
Version 2.0
Statistical Reasoning
Suggested Instructor Notes 12.D, Probability and critical values (continued)
The Charles A. Dana Center at
The University of Texas at Austin
32
Version 2.0
Statistical Reasoning
Practice Assignment 12.D
Practice Assignment 12.D
1)
The heights of American men are approximately normally distributed with a mean
of 69 inches and a standard deviation of 2.5 inches.
Part A: If the design team decides to make the car comfortable for all but the tallest
2.5% of males, what would be the cutoff height?
Part B: If the design team also decides to make the car comfortable for all but the
shortest 2.5% of males, what would be the cutoff height?
Part C: What proportion of American men are more than 5 feet tall?
Part D: What proportion of American men are more than 6 feet tall?
33
Version 2.0
Statistical Reasoning
Practice Assignment 12.D
Part E: What is the cutoff height for the tallest 10% of American men?
Part F: What are the cutoff heights for the middle 70% of American men?
2)
A random variable (X) has a continuous normal probability distribution with µ = 300
and σ = 40. Complete the table. The first row has been completed for you.
x
z
P(X < x)
P(X > x)
380
2.00
.977
.023
.691
.003
.767
.500
P(X < x) means "the probability that a given value of X is less than the value in the table."
P(X > x) means "the probability that a given value of X is more than the value in the table."
[continued on next page]
34
Version 2.0
Statistical Reasoning
Practice Assignment 12.D
3)
A random variable (X) has a continuous normal probability distribution with µ = 4.2
and σ = .89. Complete the table. The first row has been completed for you.
x
z
P(X < x)
P(X > x)
3.0
–1.35
.089
.911
.036
.184
.978
.022
.156
P(X < x) means "the probability that a given value of X is less than the value in the table."
P(X > x) means "the probability that a given value of X is more than the value in the table."
4)
A company regularly chooses 100 people to survey through random selection.
They want to ensure that their survey groups are reasonably representative of the
U.S. population. 17% of the U.S. population is Hispanic. If the company truly used
simple random sampling of the U.S. population, the number of Hispanics in the
survey groups would follow the probability distribution shown below.
These data look roughly normal, so we can approximate this with a normal
distribution with the same mean and standard deviation.
35
Version 2.0
Statistical Reasoning
Practice Assignment 12.D
Part A: An analyst wants to ensure that there are enough Hispanics in each survey
group to be representative of the U.S. population. He suggests that if the number
of Hispanics in a given survey group is in the bottom 5% of this distribution, then
it's likely that there was bias in how the survey group was chosen. Therefore, the
survey group should be discarded. Complete the following sentence.
The analyst should discard survey groups that have fewer than ________
Hispanics.
Part B: Another analyst suggests that too many Hispanics may also be a sign of
bias. She would like to keep only the survey groups that are in the middle 95% of
the distribution. Complete the following sentence.
The analyst should discard survey groups that have fewer than ______ Hispanics
and more than _____ Hispanics.
36
Version 2.0