CHAPTER ELEVEN
COMPARISONS INVOLVING PROPORTIONS
AND A TEST OF INDEPENDENCE
MULTIPLE CHOICE QUESTIONS
In the following multiple-choice questions, circle the correct answer.
1. The test of independence presented in our textbook requires that there be
a. two variables, each having two outcomes
b. two variables, each having two or more outcomes
c. two or more variables, each having two outcomes
d. two or more variables, each having two or more outcomes
2. The properties of a multinomial experiment include all of the following except
a. the experiment consists of a sequence of n identical trials
b. three or more outcomes are possible on each trial
c. the probability of each outcome can change from trial to trial
d. the trials are independent
3. In the case of the test of independence, the number of degrees of freedom for the
appropriate chi-square distribution is computed as
a. k − 1
b. k − 2
c. (r – 1)(c – 1)
d. rc − 2
4. In conducting a hypothesis test about p1 – p2, any of the following approaches can be
used except
a. comparing the observed frequencies to the expected frequencies
b. comparing the p-value to
c. comparing the hypothesized difference to the confidence interval
d. comparing the test statistic to the critical value
5. The test statistic for the chi-square tests in our textbook requires, for each category, an
expected frequency of at least
a. 2
b. 5
c. 10
d. 30
6. Both the hypothesis test for proportions of a multinomial population and the test of
independence employ the
a. F distribution
b. t distribution
c. normal distribution
d. chi-square distribution
7. The test for goodness of fit
a. is always a one-tail test with the rejection region occurring in the upper tail
b. is always a one-tail test with the rejection region occurring in the lower tail
c. is always a two-tail test
d. can be a one-tail or two-tail test
8. The assumptions for the multinomial experiment parallel those for the binomial
experiment with the exception that for the multinomial
a. there are more trials
b. the probability of each outcome can change from trial to trial
c. there are three or more outcomes per trial
d. the trials are not independent
9. Both the hypothesis test for proportions of a multinomial population and the test of
independence focus on the difference between
a. sample means and population means
b. observed frequencies and expected frequencies
c. two population proportions
d. two interval estimates
10. The purpose of the hypothesis test for proportions of a multinomial population is to
determine whether the actual proportions
a. are all equal
b. follow a normal distribution
c. are different than the hypothesized proportions
d. follow a chi-square distribution
11. If we are interested in testing whether the proportion of items in population 1 is larger
than the proportion of items in population 2, the
a. null hypothesis should state p1 − p2 > 0
b. null hypothesis should state p1 − p2 0
c. alternative hypothesis should state p1 − p2 0
d. alternative hypothesis should state p1 − p2 0
12. Assume we are interested in determining whether the proportion of voters planning to
vote for candidate C (pC) is significantly less than the proportion of voters planning to
vote for candidate B (pB). The correct null hypothesis for testing the above is
a. Ho: pC − pB 0
b. Ho: pC − pB < 0
c. Ho: pC − pB 0
d. Ho: pC − pB 0
13. The sampling distribution of
−
12
pp
is approximated by a
a. normal distribution
b. t distribution with n1 + n2 degrees of freedom
c. t distribution with n1 + n2 – 1 degrees of freedom
d. t distribution with n1 + n2 + 2 degrees of freedom
14. If a hypothesis is rejected at 95% confidence,
a. it must also be rejected at the 99% confidence
b. it must also be rejected at the 90% confidence
c. it will sometimes be rejected and sometimes not be rejected at the 90%
confidence
d. Not enough information is given to answer this question.
15. If the p-value is less than ,
a. the alternative hypothesis is rejected
b. the null hypothesis is rejected
c. the null hypothesis will sometimes be rejected and sometimes not be rejected
depending on the sample size
d. Not enough information is given to answer this question.
16. In a two-tailed hypothesis test the test statistic is determined to be z = -2.5. The p-value
for this test
a. is 0.0062
b. is 0.0124
c. is 0.4938
d. cannot be determined, since the level of confidence is not given.
17. Which of the following does not need to be known in order to compute the p-value?
a. knowledge of whether the test is one-tailed or two-tailed
b. the value of the test statistic
c. the level of significance
d. All of the information provided is necessary.
18. A population where each element of the population is assigned to one and only one of
several classes or categories is a
a. multinomial population
b. Poisson population
c. normal population
d. None of these alternatives is correct.
19. The sampling distribution for a goodness of fit test is the
a. Poisson distribution
b. t distribution
c. normal distribution
d. chi-square distribution
20. A goodness of fit test is always conducted as a
a. lower-tail test
b. upper-tail test
c. middle test
d. None of these alternatives is correct.
21. An important application of the chi-square distribution is
a. making inferences about a single population variance
b. testing for goodness of fit
c. testing for the independence of two variables
d. All of these alternatives are correct.
22. The number of degrees of freedom for the appropriate chi-square distribution in a test of
independence is
a. n − 1
b. k − 1
c. number of rows minus 1 times number of columns minus 1
d. a chi-square distribution is not used
23. In order not to violate the requirements necessary to use the chi-square distribution, each
expected frequency in a goodness of fit test must be
a. at least 5
b. at least 10
c. no more than 5
d. less than 2
24. A statistical test conducted to determine whether to reject or not reject a hypothesized
probability distribution for a population is known as a
a. contingency test
b. probability test
c. goodness of fit test
d. None of these alternatives is correct.
25. The degrees of freedom for a contingency table with 12 rows and 12 columns is
a. 144
b. 121
c. 12
d. 120
26. The degrees of freedom for a contingency table with 6 rows and 3 columns is
a. 18
b. 15
c. 6
d. 10
27. The degrees of freedom for a contingency table with 10 rows and 11 columns is
a. 100
b. 110
c. 21
d. 90
Exhibit 11-1
The results of a recent poll on the preference of shoppers regarding two products are shown
below.
Shoppers Favoring
Product
Shoppers Surveyed
This Product
A
800
560
B
900
612
28. Refer to Exhibit 11-1. The point estimate for the difference between the two population
proportions in favor of this product is
a. 52
b. 100
c. 0.44
d. 0.02
29. Refer to Exhibit 11-1. The standard error of
−
12
pp
is
a. 52
b. 0.044
c. 0.0225
d. 100
30. Refer to Exhibit 11-1. At 95% confidence, the margin of error is
a. 0.064
b. 0.044
c. 0.0225
d. 52
31. Refer to Exhibit 11-1. The 95% confidence interval estimate for the difference between
the populations favoring the products is
a. -0.024 to 0.064
b. 0.6 to 0.7
c. 0.024 to 0.7
d. 0.02 to 0.3
Exhibit 11-2
An insurance company selected samples of clients under 18 years of age and over 18 and
recorded the number of accidents they had in the previous year. The results are shown below.
Under Age of 18
Over Age of 18
n1 = 500
n2 = 600
Number of accidents = 180
Number of accidents = 150
We are interested in determining if the accident proportions differ between the two age
groups.
32. Refer to Exhibit 11-2 and let pU represent the proportion under and pO the proportion over
the age of 18. The null hypothesis is
a. pU − pO 0
b. pU − pO 0
c. pU − pO 0
d. pU − pO = 0
33. Refer to Exhibit 11-2. The pooled proportion is
a. 0.305
b. 0.300
c. 0.027
d. 0.450
34. Refer to Exhibit 11-2. The test statistic is
a. 0.96
b. 1.96
c. 2.96
d. 3.96
35. Refer to Exhibit 11-2. The p-value is
a. less than 0.001
b. more than 0.10
c. 0.0228
d. 0.3
Exhibit 11-3
The results of a recent poll on the preference of teenagers regarding the types of music they listen
to are shown below.
Music
Type
Teenagers
Surveyed
Teenagers Favoring
This Type
Pop
800
384
Rap
900
450
36. Refer to Exhibit 11-3. The point estimate for the difference between the proportions is
a. -0.02
b. 0.048
c. 100
d. 66
37. Refer to Exhibit 11-3. The standard error of
−
12
pp
is
a. 0.48
b. 0.50
c. 0.03
d. 0.0243
38. Refer to Exhibit 11-3. The 95% confidence interval for the difference between the two
proportions is
a. 384 to 450
b. 0.48 to 0.5
c. 0.028 to 0.068
d. -0.068 to 0.028
Exhibit 11-4
When individuals in a sample of 150 were asked whether or not they supported capital
punishment, the following information was obtained.
Do you support
Number of
capital punishment?
individuals
Yes
40
No
60
No Opinion
50
We are interested in determining whether or not the opinions of the individuals (as to Yes, No,
and No Opinion) are uniformly distributed.
39. Refer to Exhibit 11-4. The expected frequency for each group is
a. 0.333
b. 0.50
c. 1/3
d. 50
40. Refer to Exhibit 11-4. The calculated value for the test statistic equals
a. 2
b. -2
c. 20
d. 4
41. Refer to Exhibit 11-4. The number of degrees of freedom associated with this problem is
a. 150
b. 149
c. 2
d. 3
42. Refer to Exhibit 11-4. The p-value is
a. larger than 0.1
b. less than 0.1
c. less than 0.05
d. larger than 0.9
43. Refer to Exhibit 11-4. The conclusion of the test (at 95% confidence) is that the
a. distribution is uniform
b. distribution is not uniform
c. test is inconclusive
d. None of these alternatives is correct.
Exhibit 11-5
Last school year, the student body of a local university consisted of 30% freshmen, 24%
sophomores, 26% juniors, and 20% seniors. A sample of 300 students taken from this year’s
student body showed the following number of students in each classification.
Freshmen
83
Sophomores
68
Juniors
85
Seniors
64
We are interested in determining whether or not there has been a significant change in the
classifications between the last school year and this school year.
44. Refer to Exhibit 11-5. The expected number of freshmen is
a. 83
b. 90
c. 30
d. 10
45. Refer to Exhibit 11-5. The expected frequency of seniors is
a. 60
b. 20%
c. 68
d. 64
46. Refer to Exhibit 11-5. The calculated value for the test statistic equals
a. 0.5444
b. 300
c. 1.6615
d. 6.6615
47. Refer to Exhibit 11-5. The p-value is
a. less than .005
b. between .025 and 0.05
c. between .05 and 0.1
d. greater than 0.1
48. Refer to Exhibit 11-5. At 95% confidence, the null hypothesis
a. should not be rejected
b. should be rejected
c. was designed wrong
Exhibit 11-6
In order to determine whether or not a particular medication was effective in curing the common
cold, one group of patients was given the medication, while another group received sugar pills.
The results of the study are shown below.
Patients Cured
Patients Not Cured
Received medication
70
10
Received sugar pills
20
50
We are interested in determining whether or not the medication was effective in curing the
common cold.
49. Refer to Exhibit 11-6. The expected frequency of those who received medication and
were cured is
a. 70
b. 150
c. 28
d. 48
50. Refer to Exhibit 11-6. The test statistic is
a. 10.08
b. 54.02
c. 1.96
d. 1.645
51. Refer to Exhibit 11-6. The number of degrees of freedom associated with this problem is
a. 4
b. 149
c. 1
d. 3
52. Refer to Exhibit 11-6. The hypothesis is to be tested at the 5% level of significance. The
critical value from the table equals
a. 3.84
b. 7.81
c. 5.99
d. 9.34
53. Refer to Exhibit 11-6. The p-value is
a. less than .005
b. between .005 and .01
c. between .01 and .025
d. between .025 and .05
Exhibit 11-7
In the past, 35% of the students at ABC University were in the Business College, 35% of the
students were in the Liberal Arts College, and 30% of the students were in the Education College.
To see whether or not the proportions have changed, a sample of 300 students was taken. Ninety
of the sample students are in the Business College, 120 are in the Liberal Arts College, and 90 are
in the Education College.
54. Refer to Exhibit 11-7. This problem is an example of a
a. normally distributed variable
b. test for independence
c. Poisson distributed variable
d. multinomial population
55. Refer to Exhibit 11-7. The expected frequency for the Business College is
a. 0.3
b. 0.35
c. 90
d. 105
56. Refer to Exhibit 11-7. The calculated value for the test statistic equals
a. 0.01
b. 0.75
c. 4.29
d. 4.38
57. Refer to Exhibit 11-7. The hypothesis is to be tested at the 5% level of significance. The
critical value from the table equals
a. 1.645
b. 1.96
c. 5.991
d. 7.815
58. Refer to Exhibit 11-7. The p-value is
a. greater than 0.1
b. between 0.05 and 0.1
c. between 0.025 and 0.05
d. between 0.01 and .025
59. Refer to Exhibit 11-7. The conclusion of the test is that the
a. proportions have changed significantly
b. proportions have not changed significantly
c. test is inconclusive
d. None of these alternatives is correct.
Exhibit 11-8
The table below gives beverage preferences for random samples of teens and adults.
Teens
Adults
Total
Coffee
50
200
250
Tea
100
150
250
Soft Drink
200
200
400
Other
50
50
100
400
600
1,000
We are asked to test for independence between age (i.e., adult and teen) and drink preferences.
60. Refer to Exhibit 11-8. With a .05 level of significance, the critical value for the test is
a. 1.645
b. 7.815
c. 14.067
d. 15.507
61. Refer to Exhibit 11-8. The expected number of adults who prefer coffee is
a. 0.25
b. 0.33
c. 150
d. 200
62. Refer to Exhibit 11-8. The test statistic for this test of independence is
a. 0
b. 8.4
c. 62.5
d. 82.5
63. Refer to Exhibit 11-8. The p-value is
a. between .1 and .05
b. between .05 and .025
c. between .025 and .01
d. less than 0.005
PROBLEMS
1. 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-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.
2. 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
proportions 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.
3. Among a sample of 50 MDs (medical doctors) in the city of Memphis, Tennessee, 10
indicated they make house calls; while among a sample of 100 MDs in Atlanta, Georgia,
18 said they make house calls. Determine a 95% interval estimate for the difference
between the proportions of doctors who make house calls in the two cities.
4. 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
proportions 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?
5. During the primary elections of 2004, 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.
6. 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.
7. 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
proportions of voters favoring each candidate.
b. Does your confidence interval provide conclusive evidence that one of the candidates
is favored more? Explain.
8. 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.
9. 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 proportions of defectives in
the two lines.
10. A poll was taken this year asking college students if they considered themselves
overweight. A similar poll was taken five ago. Results are summarized below. Has the
proportion increased significantly? Let = 0.05.
Sample Size
Number Considered
Themselves Overweight
Current Sample (c)
300
150
Previous Sample (p)
275
121
11. 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.)
12. 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.
13. A comparative study of organic and conventionally grown produce checked for the
presence of E. coli. Results are summarized below. Is there a significant difference in
the proportion of E. Coli in organic versus conventionally grown produce? Test at =
0.10.
Sample Size
E. Coli Prevalence
Organic
200
3
Conventional
500
20
14. 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 Democrats and the Republicans
who favored the President’s anti-drug program.
15. 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 (p)
250
75
Current Sample (c)
300
69
16. 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.
17. 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.
18. A school administrator believes that there is no difference between student dropout rate
for schools located in rural areas and schools located in urban areas. A random sample of
100 schools in the rural areas was taken. The student dropout rate of the schools in the
sample was 27%. A random sample of 80 schools in the urban areas had a dropout rate
of 20%.
a. Give a point estimate for the difference between the population proportions for the
two districts.
b. Give a point estimate of the standard deviation for the difference between the
population proportions.
c. Compute the test statistic for testing the administrator’s belief.
d. At 95% confidence using the p-value approach, test the administrator’s belief.
19. Before the start of the Winter Olympics, it was expected that the percentages of medals
awarded to the top contenders to be as follows.
Percentages
United States
25%
Germany
22%
Norway
18%
Austria
14%
Russia
11%
France
10%
Midway through the Olympics, of the 120 medals awarded, the following distribution
was observed.
Number of Medals
United States
33
Germany
36
Norway
18
Austria
15
Russia
12
France
6
We want to test to see if there is a significant difference between the expected and actual
awards given.
a. Compute the test statistic.
b. Using the p-value approach, test to see if there is a significant difference between the
expected and the actual values. Let = .05.
c. At 95% confidence, test for a significant difference using the critical value approach.
20. A medical journal reported the following frequencies of deaths due to cardiac arrest for
each day of the week.
Day
Cardiac Deaths
Monday
40
Tuesday
17
Wednesday
16
Thursday
29
Friday
15
Saturday
20
Sunday
17
We want to determine whether the number of deaths is uniform over the week.
a. Compute the test statistic.
b. Using the p-value approach at 95% confidence, test for the uniformity of death over
the week.
c. Using the critical value approach, perform the test for uniformity.
21. Before the presidential debates, it was expected that the percentages of registered voters
in favor of various candidates would be as follows.
Percentage
Democrats
48%
Republicans
38%
Independents
4%