140
married
238
widowed
2
divorced/separated
20
Table 11.19
The next two questions refer to the following information. The columns in the chart below contain the
Race/Ethnicity of U.S. Public Schools for a recent year, the percentages for the Advanced Placement Exami-
nee Population for that class and the Overall Student Population. (Source: http://www.collegeboard.com).
Suppose the right column contains the result of a survey of 1000 local students from that year who took an
AP Exam.
Race/Ethnicity
AP Examinee Popula-
Overall Student Popu-
Survey Frequency
tion
lation
Asian, Asian American
10.2%
5.4%
113
or Pacific Islander
Black or African Ameri-
8.2%
14.5%
94
can
Hispanic or Latino
15.5%
15.9%
136
American
Indian
or
0.6%
1.2%
10
Alaska Native
White
59.4%
61.6%
604
Not reported/other
6.1%
1.4%
43
Table 11.20
Exercise 11.13.4
Perform a goodness-of-fit test to determine whether the local results follow the distribution of the
U. S. Overall Student Population based on ethnicity.
Exercise 11.13.5
(Solution on p. 518.)
Perform a goodness-of-fit test to determine whether the local results follow the distribution of U.
S. AP Examinee Population, based on ethnicity.
Exercise 11.13.6
The City of South Lake Tahoe, CA, has an Asian population of 1419 people, out of a total popu-
lation of 23,609 (Source: U.S. Census Bureau). Suppose that a survey of 1419 self-reported Asians
in Manhattan, NY, area yielded the data in the table below. Conduct a goodness of fit test to de-
termine if the self-reported sub-groups of Asians in the Manhattan area fit that of the Lake Tahoe
area.
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Race
Lake Tahoe Frequency
Manhattan Frequency
Asian Indian
131
174
Chinese
118
557
Filipino
1045
518
Japanese
80
54
Korean
12
29
Vietnamese
9
21
Other
24
66
Table 11.21
The next two questions refer to the following information: UCLA conducted a survey of more than
263,000 college freshmen from 385 colleges in fall 2005. The results of student expected majors by gender
were reported in The Chronicle of Higher Education (2/2/2006). Suppose a survey of 5000 graduating
females and 5000 graduating males was done as a follow-up last year to determine what their actual major
was. The results are shown in the tables for Exercises 7 and 8. The second column in each table does not
add to 100% because of rounding.
Exercise 11.13.7
(Solution on p. 519.)
Conduct a hypothesis test to determine if the actual college major of graduating females fits the
distribution of their expected majors.
Major
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 11.22
Exercise 11.13.8
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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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 11.23
Exercise 11.13.9
(Solution on p. 519.)
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 11.24
Exercise 11.13.10
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 11.25
Exercise 11.13.11
(Solution on p. 519.)
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
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CHAPTER 11. THE CHI-SQUARE DISTRIBUTION
in college and their starting salaries after graduation. Below are the data. Conduct a test for
independence.
Major
< $50,000
$50,000 - $68,999
$69,000 +
English
5
20
5
Engineering
10
30
60
Nursing
10
15
15
Business
10
20
30
Psychology
20
30
20
Table 11.26
Exercise 11.13.12
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 11.27
Exercise 11.13.13
(Solution on p. 519.)
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 11.28
Exercise 11.13.14
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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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 11.29
Exercise 11.13.15
(Solution on p. 519.)
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
< $200,000
$200,000 - $400,000
$401,001 - $1,000,000
$1,000,000 +
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 11.30
Exercise 11.13.16
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-
graduate
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 11.31
Exercise 11.13.17
(Solution on p. 519.)
A Psychologist is interested in testing whether there is a difference in the distribution of personal-
ity types for business majors and social science majors. The results of the study are shown below.
Conduct a Test of Homogeneity. Test at a 5% level of significance.
Open
Conscientious
Extrovert
Agreeable Neurotic
Business
41
52
46
61 58
Social Science
72
75
63
80 65
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CHAPTER 11. THE CHI-SQUARE DISTRIBUTION
Table 11.32
Exercise 11.13.18
(Solution on p. 519.)
Do men and women select different breakfasts? The breakfast ordered by randomly selected men
and women at a popular breakfast place is shown below. Conduct a test of homogeneity. Test at a
5% level of significance
French Toast
Pancakes
Waffles
Omelettes
Men
47
35
28
53
Women
65
59
55
60
Table 11.33
Exercise 11.13.19
(Solution on p. 520.)
Is there a difference between the distribution of community college statistics students and the
distribution of university statistics students in what technology they use on their homework? Of
the randomly selected community college students 43 used a computer, 102 used a calculator
with built in statistics functions, and 65 used a table from the textbook. Of the randomly selected
university students 28 used a computer, 33 used a calculator with built in statistics functions, and
40 used a table from the textbook. Conduct an appropriate hypothesis test using a 0.05 level of
significance.
Exercise 11.13.20
(Solution on p. 520.)
A fisherman is interested in whether the distribution of fish caught in Green Valley Lake is the
same as the distribution of fish caught in Echo Lake. Of the 191 randomly selected fish caught in
Green Valley Lake, 105 were rainbow trout, 27 were other trout, 35 were bass, and 24 were catfish.
Of the 293 randomly selected fish caught in Echo Lake, 115 were rainbow trout, 58 were other
trout, 67 were bass, and 53 were catfish. Perform the hypothesis test at a 5% level of significance.
Exercise 11.13.21
(Solution on p. 520.)
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 11.13.22
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.
Exercise 11.13.23
(Solution on p. 520.)
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 11.13.24
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.
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Exercise 11.13.25
(Solution on p. 520.)
The number of births per woman in China is 1.6 down from 5.91 in 1966 (SourceWorld Bank,
6/5/12 ). 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 11.34
Exercise 11.13.26
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 11.13.27
(Solution on p. 520.)
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 a mean of 11 oz. and a standard deviation of 2 oz.
11.13.2 Try these true/false questions.
Exercise 11.13.28
(Solution on p. 521.)
As the degrees of freedom increase, the graph of the chi-square distribution looks more and more
symmetrical.
Exercise 11.13.29
(Solution on p. 521.)
The standard deviation of the chi-square distribution is twice the mean.
Exercise 11.13.30
(Solution on p. 521.)
The mean and the median of the chi-square distribution are the same if df = 24.
Exercise 11.13.31
(Solution on p. 521.)
In a Goodness-of-Fit test, the expected values are the values we would expect if the null hypoth-
esis were true.
Exercise 11.13.32
(Solution on p. 521.)
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 11.13.33
(Solution on p. 521.)
The degrees of freedom for a Test for Independence are equal to the sample size minus 1.
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CHAPTER 11. THE CHI-SQUARE DISTRIBUTION
Exercise 11.13.34
(Solution on p. 521.)
Use a Goodness-of-Fit test to determine if high school principals believe that students are absent
equally during the week or not.
Exercise 11.13.35
(Solution on p. 521.)
The Test for Independence uses tables of observed and expected data values.
Exercise 11.13.36
(Solution on p. 521.)
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 11.13.37
(Solution on p. 521.)
The test to use to determine if a six-sided die is fair is a Goodness-of-Fit test.
Exercise 11.13.38
(Solution on p. 521.)
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 11.13.39
(Solution on p. 521.)
In a Goodness-of Fit test, if the p-value is 0.0113, in general, do not reject the null hypothesis.
Exercise 11.13.40
(Solution on p. 521.)
For a Chi-Square distribution with degrees of freedom of 17, the probability that a value is greater
than 20 is 0.7258.
Exercise