Women - Expected Major
Women - Actual Major
Arts & Humanities
14.0%
670
Biological Sciences
8.4%
410
Business
13.1%
685
Education
13.0%
650
Engineering
2.6%
145
Physical Sciences
2.6%
125
Professional
18.9%
975
Social Sciences
13.0%
605
Technical
0.4%
15
Other
5.8%
300
Undecided
8.0%
420
Table 11.20
Exercise 11.9.8
Conduct a hypothesis test to determine if the actual college major of graduating males fits the
distribution of their expected majors.
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CHAPTER 11. THE CHI-SQUARE DISTRIBUTION
Major
Men - Expected Major
Men - Actual Major
Arts & Humanities
11.0%
600
Biological Sciences
6.7%
330
Business
22.7%
1130
Education
5.8%
305
Engineering
15.6%
800
Physical Sciences
3.6%
175
Professional
9.3%
460
Social Sciences
7.6%
370
Technical
1.8%
90
Other
8.2%
400
Undecided
6.6%
340
Table 11.21
Exercise 11.9.9
(Solution on p. 500.)
A recent debate about where in the United States skiers believe the skiing is best prompted the
following survey. Test to see if the best ski area is independent of the level of the skier.
U.S. Ski Area
Beginner
Intermediate
Advanced
Tahoe
20
30
40
Utah
10
30
60
Colorado
10
40
50
Table 11.22
Exercise 11.9.10
Car manufacturers are interested in whether there is a relationship between the size of car an
individual drives and the number of people in the driver’s family (that is, whether car size and
family size are independent). To test this, suppose that 800 car owners were randomly surveyed
with the following results. Conduct a test for independence.
Family Size
Sub & Compact
Mid-size
Full-size
Van & Truck
1
20
35
40
35
2
20
50
70
80
3 - 4
20
50
100
90
5+
20
30
70
70
Table 11.23
Exercise 11.9.11
(Solution on p. 501.)
College students may be interested in whether or not their majors have any effect on starting
salaries after graduation. Suppose that 300 recent graduates were surveyed as to their majors
485
in college and their starting salaries after graduation. Below are the data. Conduct a test for
independence.
Major
< $30,000
$30,000 - $39,999
$40,000 +
English
5
20
5
Engineering
10
30
60
Nursing
10
15
15
Business
10
20
30
Psychology
20
30
20
Table 11.24
Exercise 11.9.12
Some travel agents claim that honeymoon hot spots vary according to age of the bride and groom.
Suppose that 280 East Coast recent brides were interviewed as to where they spent their honey-
moons. The information is given below. Conduct a test for independence.
Location
20 - 29
30 - 39
40 - 49
50 and over
Niagara Falls
15
25
25
20
Poconos
15
25
25
10
Europe
10
25
15
5
Virgin Islands
20
25
15
5
Table 11.25
Exercise 11.9.13
(Solution on p. 501.)
A manager of a sports club keeps information concerning the main sport in which members
participate and their ages. To test whether there is a relationship between the age of a member
and his or her choice of sport, 643 members of the sports club are randomly selected. Conduct a
test for independence.
Sport
18 - 25
26 - 30
31 - 40
41 and over
racquetball
42
58
30
46
tennis
58
76
38
65
swimming
72
60
65
33
Table 11.26
Exercise 11.9.14
A major food manufacturer is concerned that the sales for its skinny French fries have been de-
creasing. As a part of a feasibility study, the company conducts research into the types of fries sold
across the country to determine if the type of fries sold is independent of the area of the country.
The results of the study are below. Conduct a test for independence.
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CHAPTER 11. THE CHI-SQUARE DISTRIBUTION
Type of Fries
Northeast
South
Central
West
skinny fries
70
50
20
25
curly fries
100
60
15
30
steak fries
20
40
10
10
Table 11.27
Exercise 11.9.15
(Solution on p. 501.)
According to Dan Lenard, an independent insurance agent in the Buffalo, N.Y. area, the following
is a breakdown of the amount of life insurance purchased by males in the following age groups.
He is interested in whether the age of the male and the amount of life insurance purchased are
independent events. Conduct a test for independence.
Age of Males
None
$50,000 - $100,000
$100,001 - $150,000
$150,001 - $200,000
$200,000 +
20 - 29
40
15
40
0
5
30 - 39
35
5
20
20
10
40 - 49
20
0
30
0
30
50 +
40
30
15
15
10
Table 11.28
Exercise 11.9.16
Suppose that 600 thirty–year–olds were surveyed to determine whether or not there is a relation-
ship between the level of education an individual has and salary. Conduct a test for independence.
Annual Salary
Not a high school
High school grad-
College graduate
Masters or doctor-
grad.
uate
ate
< $30,000
15
25
10
5
$30,000 - $40,000
20
40
70
30
$40,000 - $50,000
10
20
40
55
$50,000 - $60,000
5
10
20
60
$60,000 +
0
5
10
150
Table 11.29
Exercise 11.9.17
(Solution on p. 501.)
A plant manager is concerned her equipment may need recalibrating. It seems that the actual
weight of the 15 oz. cereal boxes it fills has been fluctuating. The standard deviation should be
at most 1 oz. In order to determine if the machine needs to be recalibrated, 84 randomly selected
2
boxes of cereal from the next day’s production were weighed. The standard deviation of the 84
boxes was 0.54. Does the machine need to be recalibrated?
Exercise 11.9.18
Consumers may be interested in whether the cost of a particular calculator varies from store to
store. Based on surveying 43 stores, which yielded a sample mean of $84 and a sample standard
deviation of $12, test the claim that the standard deviation is greater than $15.
487
Exercise 11.9.19
(Solution on p. 501.)
Isabella, an accomplished Bay to Breakers runner, claims that the standard deviation for her time
to run the 7 ½ mile race is at most 3 minutes. To test her claim, Rupinder looks up 5 of her race
times. They are 55 minutes, 61 minutes, 58 minutes, 63 minutes, and 57 minutes.
Exercise 11.9.20
Airline companies are interested in the consistency of the number of babies on each flight, so that
they have adequate safety equipment. They are also interested in the variation of the number of
babies. Suppose that an airline executive believes the average number of babies on flights is 6 with
a variance of 9 at most. The airline conducts a survey. The results of the 18 flights surveyed give
a sample average of 6.4 with a sample standard deviation of 3.9. Conduct a hypothesis test of the
airline executive’s belief.
Exercise 11.9.21
(Solution on p. 501.)
According to the U.S. Bureau of the Census, United Nations, in 1994 the number of births per
woman in China was 1.8. This fertility rate has been attributed to the law passed in 1979 restricting
births to one per woman. Suppose that a group of students studied whether or not the standard
deviation of births per woman was greater than 0.75. They asked 50 women across China the
number of births they had. Below are the results. Does the students’ survey indicate that the
standard deviation is greater than 0.75?
# of births
Frequency
0
5
1
30
2
10
3
5
Table 11.30
Exercise 11.9.22
According to an avid aquariest, the average number of fish in a 20–gallon tank is 10, with a
standard deviation of 2. His friend, also an aquariest, does not believe that the standard deviation
is 2. She counts the number of fish in 15 other 20–gallon tanks. Based on the results that follow, do
you think that the standard deviation is different from 2? Data: 11; 10; 9; 10; 10; 11; 11; 10; 12; 9; 7;
9; 11; 10; 11
Exercise 11.9.23
(Solution on p. 501.)
The manager of "Frenchies" is concerned that patrons are not consistently receiving the same
amount of French fries with each order. The chef claims that the standard deviation for a 10–
ounce order of fries is at most 1.5 oz., but the manager thinks that it may be higher. He randomly
weighs 49 orders of fries, which yields: mean of 11 oz., standard deviation of 2 oz.
11.9.2 Try these true/false questions.
Exercise 11.9.24
(Solution on p. 502.)
As the degrees of freedom increase, the graph of the chi-square distribution looks more and more
symmetrical.
Exercise 11.9.25
(Solution on p. 502.)
The standard deviation of the chi-square distribution is twice the mean.
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CHAPTER 11. THE CHI-SQUARE DISTRIBUTION
Exercise 11.9.26
(Solution on p. 502.)
The mean and the median of the chi-square distribution are the same if df = 24.
Exercise 11.9.27
(Solution on p. 502.)
In a Goodness-of-Fit test, the expected values are the values we would expect if the null hypoth-
esis were true.
Exercise 11.9.28
(Solution on p. 502.)
In general, if the observed values and expected values of a Goodness-of-Fit test are not close
together, then the test statistic can get very large and on a graph will be way out in the right tail.
Exercise 11.9.29
(Solution on p. 502.)
The degrees of freedom for a Test for Independence are equal to the sample size minus 1.
Exercise 11.9.30
(Solution on p. 502.)
Use a Goodness-of-Fit test to determine if high school principals believe that students are absent
equally during the week or not.
Exercise 11.9.31
(Solution on p. 502.)
The Test for Independence uses tables of observed and expected data values.
Exercise 11.9.32
(Solution on p. 502.)
The test to use when determining if the college or university a student chooses to attend is related
to his/her socioeconomic status is a Test for Independence.
Exercise 11.9.33
(Solution on p. 502.)
The test to use to determine if a six-sided die is fair is a Goodness-of-Fit test.
Exercise 11.9.34
(Solution on p. 502.)
In a Test of Independence, the expected number is equal to the row total multiplied by the column
total divided by the total surveyed.
Exercise 11.9.35
(Solution on p. 502.)
In a Goodness-of Fit test, if the p-value is 0.0113, in general, do not reject the null hypothesis.
Exercise 11.9.36
(Solution on p. 502.)
For a Chi-Square distribution with degrees of freedom of 17, the probability that a value is greater
than 20 is 0.7258.
Exercise 11.9.37
(Solution on p. 502.)
If df = 2, the chi-square distribution has a shape that reminds us of the exponential.
489
11.10 Review10
The next two questions refer to the following real study:
A recent survey of U.S. teenage pregnancy was answered by 720 girls, age 12 - 19. 6% of the girls surveyed
said they have been pregnant. (Parade Magazine) We are interested in the true proportion of U.S. girls, age
12 - 19, who have been pregnant.
Exercise 11.10.1
(Solution on p. 502.)
Find the 95% confidence interval for the true proportion of U.S. girls, age 12 - 19, who have been
pregnant.
Exercise 11.10.2
(Solution on p. 502.)
The report also stated that the results of the survey are accurate to within ± 3.7% at the 95%
confidence level. Suppose that a new study is to be done. It is desired to be accurate to within 2%
of the 95% confidence level. What is the minimum number that should be surveyed?
Exercise 11.10.3
Given: X ∼ Exp 1 . Sketch the graph that depicts: P (x > 1).
3
The next four questions refer to the following information:
Suppose that the time that owners keep their cars (purchased new) is normally distributed with a mean
of 7 years and a standard deviation of 2 years. We are interested in how long an individual keeps his car
(purchased new). Our population is people who buy their cars new.
Exercise 11.10.4
(Solution on p. 502.)
60% of individuals keep their cars at most how many years?
Exercise 11.10.5
(Solution on p. 502.)
Suppose that we randomly survey one person. Find the probability that person keeps his/her car
less than 2.5 years.
Exercise 11.10.6
(Solution on p. 502.)
If we are to pick individuals 10 at a time, find the distribution for the average car length owner-
ship.
Exercise 11.10.7
(Solution on p. 502.)
If we are to pick 10 individuals, find the probability that the sum of their ownership time is more
than 55 years.
Exercise 11.10.8
(Solution on p. 502.)
For which distribution is the median not equal to the mean?
A. Uniform
B. Exponential
C. Normal
D. Student-t
Exercise 11.10.9
(Solution on p. 502.)
Compare the standard normal distribution to the student-t distribution, centered at 0. Explain
which of the following are true and which are false.
a. As the number surveyed increases, the area to the left of -1 for the student-t distribution ap-
proaches the area for the standard normal distribution.
b. As the degrees of freedom decrease, the graph of the student-t distribution looks more like the
graph of the standard normal distribution.
10This content is available online at <http://cnx.org/content/m17057/1.10/>.
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CHAPTER 11. THE CHI-SQUARE DISTRIBUTION
c. If the number surveyed is 15, the normal distribution should never be used.
The next five questions refer to the following information:
We are interested in the checking account balance of a twenty-year-old college student. We randomly
survey 16 twenty-year-old college students. We obtain a sample mean of $640 and a sample standard
deviation of $150. Let X = checking account balance of an individual twenty year old college student.
Exercise 11.10.10
Explain why we cannot determine the distribution of X.
Exercise 11.10.11
(Solution on p. 503.)
If you were to create a confidence interval or perform a hypothesis test for the population average
checking account balance of 20-year old college students, what distribution would you use?
Exercise 11.10.12
(Solution on p. 503.)
Find the 95% confidence interval for the true average checking account balance of a twenty-year-
old college student.
Exercise 11.10.13
(Solution on p. 503.)
What type of data is the balance of the checking account considered to be?
Exercise 11.10.14
(Solution on p. 503.)
What type of data is the number of 20 year olds considered to be?
Exercise 11.10.15
(Solution on p. 503.)
On average, a busy emergency room gets a patient with a shotgun wound about once per week.
We are interested in the number of patients with a shotgun wound the emergency room gets per
28 days.
a. Define the random variable X.
b. State the distribution for X.
c. Find the probability that the emergency room gets no patients with shotgun wounds in the next
28 days.
The next two questions refer to the following information:
The probability that a certain slot machine will pay back money when a quarter is inserted is 0.30 . Assume
that each play of the slot machine is independent from each other. A person puts in 15 quarters for 15 plays.
Exercise 11.10.16
(Solution on p. 503.)
Is the expected number of plays of the slot machine that will pay back money greater than, less
than or the same as the median? Explain your answer.
Exercise 11.10.17
(Solution on p. 503.)
Is it likely that exactly 8 of the 15 plays would pay back money? Justify your answer numerically.
Exercise 11.10.18
(Solution on p. 503.)
A game is played with the following rules:
• it costs $10 to enter
• a fair coin is tossed 4 times
• if you do not get 4 heads or 4 tails, you lose your $10
• if you get 4 heads or 4 tails, you get back your $10, plus $30 more
Over the long run of playing this game, what are your expected earnings?
Exercise 11.10.19
(Solution on p. 503.)
491
• The average grade on a math exam in Rachel’s class was 74, with a standard deviation of 5.
Rachel earned an 80.
• The average grade on a math exam in Becca’s class was 47, with a standard deviation of 2.
Becca earned a 51.
• The average grade on a math exam in Matt’s class was 70, with a standard deviation of 8.
Matt earned an 83.
Find whose score was the best, compared to his or her own class. Justify your answer numerically.
The next two questions refer to the following information:
A random sample