Chapter 11 – Comparisons Involving Proportions and a Test of Independence
3
60
4
50
a.
State the null and alternative hypotheses to be tested.
b.
Compute the test statistic.
c.
The null hypothesis is to be tested at the 5% level of significance. Determine the critical value
from the table.
d.
What do you conclude about the fairness of this lottery?
9.48773
1
55. The makers of Compute-All know that in the past, 40% of their sales were from people under 30 years old, 45% of
their sales were from people who are between 30 and 50 years old, and 15% of their sales were from people who are over
50 years old. A sample of 300 customers was taken to see if the market shares had changed. In the sample, 100 of the
people were under 30 years old, 150 people were between 30 and 50 years old, and 50 people were over 50 years old.
a.
State the null and alternative hypotheses to be tested.
b.
Compute the test statistic.
c.
The null hypothesis is to be tested at the 1% level of significance. Determine the critical value
from the table.
d.
What do you conclude?
5.55
9.21034
1
56. Shown below is a 3 × 2 contingency table with observed values from a sample of 1,500. At 95% confidence, test for
independence of the row and column factors.
Column Factor
Row Factor
x
y
Total
A
450
300
750
B
300
300
600
C
150
0
150
Total
900
600
1,500
1
57. Shown below is a 2 × 3 contingency table with observed values from a sample of 500. At 95% confidence, test for
independence of the row and column factors.
Column Factor
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
Row Factor
x
y
Z
A
40
50
110
B
60
100
140
Chi-square = 4.44 < 5.99; is independent
1
58. A sample of 150 individuals (males and females) was surveyed, and the individuals were asked to indicate their yearly
incomes. The results of the survey are shown below.
Income Category
Male
Female
Category 1: $20,000 up to $40,000
10
30
Category 2: $40,000 up to $60,000
35
15
Category 3: $60,000 up to $80,000
15
45
Test at α = 0.05 to determine if the yearly income is independent of the gender.
1
59. A group of 2000 individuals from 3 different cities were asked whether they owned a foreign or a domestic car. The
following contingency table shows the results of the survey.
City
Type of Car
Detroit
Atlanta
Denver
Total
Domestic
80
200
520
800
Foreign
120
600
480
1,200
Total
200
800
1,000
2,000
At α = 0.05, test to determine if the type of car purchased is independent of the city in which the purchasers live.
1
60. Dr. Ross’ diet pills are supposed to cause significant weight loss. The following table shows the results of a recent
study where some individuals took the diet pills and some did not.
Diet Pills
No Diet Pills
Total
No Weight Loss
80
20
100
Weight Loss
100
100
200
Total
180
120
300
With 95% confidence, test to see if losing weight is dependent on taking the diet pills.
Chi-square = 25 > 5.99; is dependent
1
61. Five hundred randomly selected automobile owners were questioned on the main reason they had purchased their
current automobile. The results are given below.
Main Reason Purchased
Styling
Engineering
Fuel Economy
Total
Male
70
130
150
350
Female
30
20
100
150
Total
100
150
250
500
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
a.
State the null and alternative hypotheses for a contingency table test.
b.
State the decision rule, using a .10 level of significance.
c.
Calculate the chi-square test statistic.
d.
Give your conclusion for this test.
31.746
Reject the null hypothesis and conclude automobile preference is not independent of sex
1
62. A group of 500 individuals were asked to cast their votes regarding a particular issue of the Equal Rights Amendment.
The following contingency table shows the results of the votes:
Vote Cast
Gender
Favor
Undecided
Oppose
Total
Female
180
80
40
300
Male
150
20
30
200
Total
330
100
70
500
Test at α = .05 to determine if the votes cast were independent of the gender of the individuals.
Chi-square = 20.99 > 5.99; votes are not independent of gender.
1
63. One thousand managers with degrees in business administration indicated their fields of concentration as shown
below.
Major
Top Management
Middle Management
Total
Management
300
200
500
Marketing
200
0
200
Accounting
100
200
300
Total
600
400
1,000
Test at α = .01 to determine if the position in management is independent of the major of concentration.
Chi-square = 222.2 > 9.21; the position is not independent of major
1
64. From a poll of 800 television viewers, the following data have been accumulated as to their levels of education and
their preference of television stations:
Level of Educational
High School
Bachelor
Graduate
Total
Public Broadcasting
150
150
100
400
Commercial Stations
50
250
100
400
Total
200
400
200
800
Test at α = .05 to determine if the selection of a TV station is dependent upon the level of education.
Chi-square = 75 > 5.99; selection of station is not independent of the level of education.
1
65. The data below represents the fields of specialization for a randomly selected sample of undergraduate students. Test
to determine whether there is a significant difference in the fields of specialization between regions of the country. Use a
.05 level of significance.
Region of United States
Specialization
Northeast
Midwest
South
West
Total
Business
54
65
28
93
240
Engineering
15
24
8
33
80
Liberal Arts
65
84
33
98
280
Fine Arts
13
15
7
25
60
Health Sciences
3
12
4
21
40
Total
150
200
80
270
700
a.
State the critical value of the chi-square random variable for this test of independence of
categories.
b.
Calculate the value of the test statistic.
c.
What is the conclusion for this test?
21.0261
8.674
Do not reject the null hypothesis that fields of specialization and region are independent.
66. During “sweeps week” last year, the viewing audience was distributed as follows: 36% NBC, 22% ABC, 24% CBS,
and 18% FOX. This year during “sweeps week” a sample of 50 homes yielded the following data. Use Excel to test at α =
.05 to determine if the audience proportions have changed.
ABC
FOX
ABC
FOX
ABC
ABC
CBS
NBC
FOX
FOX
NBC
ABC
CBS
ABC
NBC
NBC
NBC
CBS
FOX
ABC
ABC
FOX
NBC
CBS
CBS
NBC
NBC
ABC
FOX
FOX
NBC
NBC
NBC
NBC
FOX
ABC
FOX
NBC
FOX
CBS
CBS
CBS
FOX
FOX
NBC
CBS
FOX
CBS
FOX
NBC
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
67. Members of a focus group stated their preferences between three possible slogans. The results follow. Use Excel to
test at α = .05 to determine any difference in preference among the three slogans.
Slogan Preferences
A
A
C
C
B
C
B
B
A
A
B
C
A
B
C
C
C
C
B
B
C
B
C
C
A
A
A
C
A
B
changed
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
68. A study of wage discrimination at a local store compared employees’ race and their status. Partial results of the study
follow. Use Excel and test at α = .05 to determine if race is independent of status.
Employee
Race
Status
Employee
Race
Status
1
white
manager
26
non-white
associate
2
non-white
associate
27
white
district mgr.
3
white
district mgr.
28
non-white
manager
4
white
manager
29
white
associate
5
white
manager
30
non-white
district mgr.
6
non-white
associate
31
non-white
district mgr.
7
non-white
associate
32
white
district mgr.
8
white
associate
33
white
district mgr.
9
non-white
associate
34
non-white
associate
10
white
manager
35
white
district mgr.
11
non-white
manager
36
non-white
associate
12
non-white
associate
37
non-white
manager
13
white
associate
38
non-white
associate
14
non-white
associate
39
white
district mgr.
preference among the 3 slogans
1
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
15
white
district mgr.
40
non-white
associate
16
white
district mgr.
41
non-white
manager
17
non-white
associate
42
non-white
district mgr.
18
non-white
associate
43
white
manager
19
white
associate
44
white
district mgr.
20
non-white
manager
45
non-white
associate
21
white
district mgr.
46
non-white
associate
22
non-white
district mgr.
47
non-white
district mgr.
23
non-white
manager
48
white
manager
24
non-white
associate
49
non-white
manager
25
non-white
associate
50
non-white
associate
69. City planners are evaluating three proposed alternatives for relieving the growing traffic congestion on a north-south
highway in a booming city. The proposed alternatives are: (1) designate high-occupancy vehicle (HOV) lanes on the
existing highway, (2) construct a new, parallel highway, and (3) construct a light (passenger) rail system.
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
In an analysis of the three proposals, a citizen group has raised the question of whether preferences for the three
alternatives differ among residents near the highway and non-residents. A test of independence will address this question,
with the hypotheses being:
H0: Proposal preference is independent of the residency status of the individual
Ha: Proposal preference is not independent of the residency status of the individual
A simple random sample of 500 individuals has been selected. A crosstabulation of the residency statuses and proposal
preferences of the individuals sampled is shown below.
PROPOSAL
RESIDENCY STATUS
HOV Lanes
New Highway
Light Rail
Nearby Resident
110
45
70
Distant Resident
140
75
60
Conduct a test of independence using α = .05 to address the question of whether residency status is independent of the
proposal preference.
1
70. Of 200 UTC seniors surveyed, 60 were planning on attending Graduate School. At UTK, 400 seniors were surveyed
and 100 indicated that they were planning to attend Graduate School.
a.
Determine a 95% confidence interval estimate for the difference between the proportion of
seniors at the two universities that were planning to attend Graduate School.
b.
Is there conclusive evidence to prove that the proportion of students from UTC who plan to go
to Graduate School is significantly more than those from UTK? Explain.
-0.026 to 0.126
No, the range of the interval is from a negative to a positive value.
71. Of 300 female registered voters surveyed, 120 indicated they were planning to vote for the incumbent president; while
of 400 male registered voters, 140 indicated they were planning to vote for the incumbent president.
a.
Compute the test statistic.
b.
At alpha = .05, test to see if there is a significant difference between the proportions of females
and males who plan to vote for the incumbent president. (Use the p-value approach.)
Z = 1.35
72. Of 150 Chattanooga residents surveyed, 60 indicated that they participated in a recycling program. In Knoxville, 120
residents were surveyed and 36 claimed to recycle.
a.
Determine a 95% confidence interval estimate for the difference between the proportion of
residents recycling in the two cities.
b.
From your answer in Part a, is there sufficient evidence to conclude that there is a significant
difference in the proportion of residents participating in a recycling program?
-0.0134 to 0.2134
73. Among a sample of 50 M.D.’s (medical doctors) in the city of Memphis, Tennessee, 10 indicated they make house
calls; while among a sample of 100 M.D.’s in Atlanta, Georgia, 18 said they make house calls. Determine a 95% interval
estimate for the difference between the proportion of doctors who make house calls in the two cities.
-0.114 to 0.154
74. During the primary elections of 2012, candidate A showed the following pre-election voter support in Tennessee and
Mississippi.
Voters Surveyed
Voters Favoring
Candidate A
Tennessee
500
295
Mississippi
700
357
a.
Develop a 95% confidence interval estimate for the difference between the proportion of voters
favoring candidate A in the two states.
b.
Is there conclusive evidence that one of the two states had a larger proportion of voters’
support? If yes, which state? Explain.
0.023 to 0.137
support.
75. The results of a recent poll on the preference of voters regarding the presidential candidates are shown below.
Voters Surveyed
Voters Favoring
This Candidate
Candidate A
200
150
Candidate B
300
195
a.
Develop a 90% confidence interval estimate for the difference between the proportion of voters
favoring each candidate.
b.
Does your confidence interval provide conclusive evidence that one of the candidates is
favored more? Explain.
0.032 to 0.168
76. In a sample of 40 Democrats, 6 opposed the President’s foreign policy, while of 50 Republicans, 8 were opposed to his
policy. Determine a 90% confidence interval estimate for the difference between the proportions of the opinions of the
individuals in the two parties.
-0.136 to 0.116
77. In a sample of 100 Republicans, 60 favored the President’s anti-drug program. While in a sample of 150 Democrats,
84 favored his program. At 95% confidence, test to see if there is a significant difference in the proportions of the
b.
No, because the interval for the proportions ranges from a negative to a positive value.
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
Democrats and the Republicans who favored the President’s anti-drug program.
78. In a random sample of 200 Republicans, 160 opposed the new tax laws. While in a sample of 120 Democrats, 84
opposed the new tax laws. Determine a 95% confidence interval estimate for the difference between the proportions of
Republicans and Democrats opposed to this new law.
0.001 to 0.199
79. During the recent primary elections, the democratic presidential candidate showed the following pre-election voter
support in Alabama and Mississippi.
State
Voters Surveyed
Voters Favoring the
Democratic Candidate
Alabama
800
440
Mississippi
600
360
a.
We want to determine whether or not the proportions of voters favoring the Democratic
candidate were the same in both states. Provide the hypotheses.
b.
Compute the test statistic.
c.
Determine the p-value; and at 95% confidence, test the above hypotheses.
Z = -1.87
80. The office of records at a university has stated that the proportion of incoming female students who major in business
has increased. A sample of female students taken several years ago is compared with a sample of female students this
year. Results are summarized below. Has the proportion increased significantly? Test at alpha = .10.
Sample Size
No. Majoring in Business
Previous Sample
250
50
Present Sample
300
69
81. The reliability of two types of machines used in the same manufacturing process is to be tested. The first machine
failed to operate correctly in 90 out of 300 trials while the second type failed to operate correctly in 50 out of 250 trials.
a.
Give a point estimate for the difference between the population proportions of these machines.
b.
Calculate the pooled estimate of the population proportion.
c.
Carry out a hypothesis test to check whether there is a statistically significant difference in the
reliability for the two types of machines using a .10 level of significance.
82. The results of a recent poll on the preference of voters regarding presidential candidates are shown below.
Candidate
Voters
Surveyed
Voters Favoring
This Candidate
A
400
192
B
450
225
At 95% confidence, test to determine whether or not there is a significant difference between the preferences for the two
candidates.
83. From production line A, a sample of 500 items is selected at random, and it is determined that 30 items are defective.
In a sample of 300 items from production process B (which produces identical items to line A), there are 12 defective
items. Determine a 95% confidence interval estimate for the difference between the proportion of defectives in the two
lines.
-0.01 to 0.05
84. Babies weighing less than 5.5 pounds at birth are considered “low-birth-weight babies.” In the United States, 7.6% of
newborns are low-birth-weight babies. The following information was accumulated from samples of new births taken
from two counties.
Hamilton
Shelby
Sample size
150
200
Number of “low-birth-weight babies
18
22
a.
Develop a 95% confidence interval estimate for the difference between the proportions of
low-birth-weight babies in the two counties.
b.
Is there conclusive evidence that one of the proportions is significantly more than the other? If
yes, which county? Explain, using the results of part (a). Do not perform any test.
-.0577 to .0777
Since the range of the interval is from negative to positive, there is no indication that one
85. A poll was taken this year asking college students if they considered themselves overweight. A similar poll was taken
five years ago. Results are summarized below. Has the proportion increased significantly? Let α = 0.05.
Sample Size
Number Considered Themselves
Overweight
Present Sample
300
150
Previous sample
275
121
0.10
b.
0.2545
86. A comparative study of organic and conventionally grown produce was checked for the presence of E. coli. Results
are summarized below. Is there a significant difference in the proportion of E. Coli in organic vs. conventionally grown
produce? Test at α = 0.10.
Sample Size
E. Coli Prevalence
Organic
200
3
Conventional
500
20
1
1