Chapter 11 – Comparisons Involving Proportions and a Test of Independence
Multiple Choice
1. 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.
2. The chi-square test for independence involves
a.
two categorical variables
b.
two quantitative variables
c.
three or more categorical variables
d.
three or more quantitative variables
3. The number of degrees of freedom for the appropriate chi-square distribution in a test of independence is
a.
b.
c.
d.
4. 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.
5. The degrees of freedom for a contingency table with 12 rows and 12 columns is
a.
144
b.
121
c.
12
d.
120
6. The degrees of freedom for a contingency table with 6 rows and 3 columns is
a.
18
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
b.
15
c.
6
d.
10
d
1
7. The degrees of freedom for a contingency table with 10 rows and 11 columns is
a.
100
b.
110
c.
21
d.
90
d
1
8. Excel’s ____ function is used to perform a test of independence.
a.
z-Test: Two Sample for Means
b.
t-Test: Two Sample Assuming Equal Variances
c.
CHISQ.TEST
d.
NORM.S.DIST
c
1
Exhibit 11-1
When individuals in a sample of 150 were asked whether or not they supported capital punishment, the following
information was obtained.
Do You Support
Capital Punishment?
Number of
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.
9. Refer to Exhibit 11–1. The expected frequency for each group is
a.
0.333
b.
0.50
c.
1/3
d.
50
d
1
10. Refer to Exhibit 11–1. The calculated value for the test statistic equals
a.
2
b.
-2
c.
20
d.
4
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
d
1
11. Refer to Exhibit 11–1. The number of degrees of freedom associated with this problem is
a.
150
b.
149
c.
2
d.
3
c
1
12. Refer to Exhibit 11–1. The hypothesis is to be tested at the 5% level of significance. The critical value from the table
equals
a.
7.37776
b.
7.81473
c.
5.99147
d.
9.34840
c
1
13. Refer to Exhibit 11–1. The conclusion of the test is that the
a.
distribution is uniform
b.
distribution is not uniform
c.
test is inconclusive
d.
None of these alternatives is correct.
a
1
Exhibit 11-2
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.
14. Refer to Exhibit 11–2. The expected number of freshmen is
a.
83
b.
90
c.
30
d.
10
b
1
15. Refer to Exhibit 11–2. The expected frequency of seniors is
a.
60
b.
20%
c.
68
d.
64
a
16. Refer to Exhibit 11–2. The calculated value for the test statistic equals
a.
0.5444
b.
300
c.
1.6615
d.
6.6615
c
17. Refer to Exhibit 11–2. 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.
2.75
d.
7.815
18. Refer to Exhibit 11–2. The null hypothesis
a.
should not be rejected
b.
should be rejected
c.
was designed wrong
d.
None of these alternatives is correct.
a
Exhibit 11-3
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.
19. Refer to Exhibit 11–3. The expected frequency of those who received medication and were cured is
a.
70
b.
150
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
c.
28
d.
48
d
1
20. Refer to Exhibit 11–3. The test statistic is
a.
10.08
b.
54.02
c.
1.96
d.
1.645
b
1
21. Refer to Exhibit 11–3. The number of degrees of freedom associated with this problem is
a.
4
b.
149
c.
1
d.
3
c
1
22. Refer to Exhibit 11–3. 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
a
1
23. Refer to Exhibit 11–3. The null hypothesis
a.
should not be rejected
b.
should be rejected
c.
should be revised
d.
None of these alternatives is correct.
b
1
Exhibit 11-4
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.
24. Refer to Exhibit 11–4. This problem is an example of a
a.
normally distributed variable
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
b.
test for independence
c.
Poisson distributed variable
d.
multinomial population
25. Refer to Exhibit 11–4. The expected frequency for the Business College is
a.
0.3
b.
0.35
c.
90
d.
105
26. Refer to Exhibit 11–4. The calculated value for the test statistic equals
a.
0.01
b.
0.75
c.
4.29
d.
4.38
c
27. Refer to Exhibit 11–4. The hypothesis is to be tested at the 5% level of significance. The critical value from the table
equals
a.
1.645
b.
19.6
c.
5.99
d.
7.80
c
28. Refer to Exhibit 11–4. 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 11-5
The table below gives beverage preferences for random samples of teens and adults.
Beverage
Teens
Adults
Total
Coffee
50
200
250
Tea
100
150
250
Soft Drink
200
200
400
29. Refer to Exhibit 11–5. 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
b
1
30. Refer to Exhibit 11–5. The expected number of adults who prefer coffee is
a.
0.25
b.
0.33
c.
150
d.
200
c
1
31. Refer to Exhibit 11–5. The calculated value for this test for independence is
a.
0
b.
8.4
c.
62.5
d.
82.5
c
1
32. Refer to Exhibit 11–5. The result of the test is that the
a.
hypothesis of independence can be rejected
b.
hypothesis of independence cannot be rejected
c.
test is inconclusive
d.
None of these alternatives is correct.
a
1
Exhibit 11-6
The results of a recent poll on the preference of shoppers regarding two products are shown below.
Product
Shoppers Surveyed
Shoppers Favoring
This Product
A
800
560
B
900
612
33. Refer to Exhibit 11–6. The point estimate for the difference between the two population proportions in favor of this
product is
Other
400
600
1,000
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
a.
52
b.
100
c.
0.44
d.
0.02
34. Refer to Exhibit 11–6. The standard error of is
a.
52
b.
0.044
c.
0.0225
d.
100
c
35. Refer to Exhibit 11–6. At 95% confidence, the margin of error is
a.
0.064
b.
0.044
c.
0.0225
d.
52
36. Refer to Exhibit 11–6. 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
a
Exhibit 11-7
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.
37. Refer to Exhibit 11-7 and let pu represent the proportion under and po the proportion over the age of 18. The null
hypothesis is
a.
pu – po ≤ 0
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
b.
pu – po ≥ 0
c.
pu – po ≠ 0
d.
pu – po = 0
38. Refer to Exhibit 11–7. The pooled proportion is
a.
0.305
b.
0.300
c.
0.027
d.
0.450
39. Refer to Exhibit 11–7. The test statistic is
a.
0.96
b.
1.96
c.
2.96
d.
3.96
40. Refer to Exhibit 11–7. The p-value is
a.
less than 0.001
b.
more than 0.10
c.
0.0228
d.
0.3
a
Exhibit 11-8
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
41. Refer to Exhibit 11–8. The point estimate for the difference between the proportions is
a.
-0.02
b.
0.048
c.
100
d.
66
a
42. Refer to Exhibit 11–8. The standard error of is
a.
0.48
b.
0.50
c.
0.03
d.
0.0243
43. Refer to Exhibit 11–8. 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
44. 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
c
45. The sampling distribution of 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
a
Subjective Short Answer
46. Before the presidential debates, it was expected that the percentages of registered voters in favor of various candidates
to be as follows.
Percentages
Democrats
48%
Republicans
38%
Independent
4%
Chapter 11 – Comparisons Involving Proportions and a Test of Independence
Undecided
10%
After the presidential debates, a random sample of 1200 voters showed that 540 favored the Democratic candidate; 480
were in favor of the Republican candidate; 40 were in favor of the Independent candidate, and 140 were undecided. At a
5% level of significance, test to see if the proportion of voters has changed.
47. Last school year, in the school of Business Administration, 30% were Accounting majors, 24% Management majors,
26% Marketing majors, and 20% Economics majors. A sample of 300 students taken from this year’s students of the
school showed the following number of students in each major:
Accounting
83
Management
68
Marketing
85
Economics
64
Total
300
Has there been any significant change in the number of students in each major between the last school year and this school
year? Use α = 0.05.
48. A medical journal reported the following frequencies of deaths due to cardiac arrest for each day of the week:
Cardiac Death
by Day of the Week
Day
Frequency
Monday
40
Tuesday
17
Wednesday
16
Thursday
29
Friday
15
Saturday
20
Sunday
17
At a 5% level of significance, determine whether the number of deaths is uniform over the week.
49. The personnel department of a large corporation reported sixty resignations during the last year. The following table
groups these resignations according to the season in which they occurred:
Season
Resignations
Winter
10
Spring
22
Summer
19
Fall
9
Test to see if the number of resignations is uniform over the four seasons.
Let α = 0.05.
50. In 2013, forty percent of the students at a major university were Business majors, 35% were Engineering majors and
the rest of the students were majoring in other fields. In a sample of 600 students from the same university taken in 2014,
two hundred were Business majors, 220 were Engineering majors and the remaining students in the sample were majoring
in other fields. At a 5% significance level, test to see if there has been a significant change in the proportions between
2013 and 2014.
Chi-square = 13.15 > 5.99, thus, reject Ho and conclude there has been a significant change.
1
51. In the last presidential election before the candidates began their major campaigns, the percentages of registered voters
who favored the various candidates were as follows:
Registered Voters
Percentages
Republicans
34
Democrats
43
Independents
23
After the major campaigns began, a random sample of 400 voters showed that 172 favored the Republican candidate; 164
were in favor of the Democratic candidate; and 64 favored the Independent candidate. Test with α = .01 to see if the
proportion of voters who favored the various candidates changed.
Chi-square = 18.42 > 9.21; proportion has changed.
1
52. Before the rush began for Christmas shopping, a department store had noted that the percentage of its customers who
use the store’s credit card, the percentage of those who use a major credit card, and the percentage of those who pay cash
are the same. During the Christmas rush in a sample of 150 shoppers, 46 used the store’s credit card; 43 used a major
credit card; and 61 paid cash. With α = 0.05, test to see if the methods of payment have changed during the Christmas
rush.
Chi-square = 3.72 < 5.99; has not changed
1
53. A major automobile manufacturer claimed that the frequencies of repairs on all five models of its cars are the same. A
sample of 200 repair services showed the following frequencies on the various makes of cars.
Model of Car
Frequency
A
32
B
45
C
43
D
34
E
46
At α = 0.05, test the manufacturer’s claim.
Chi-square = 4.25 < 9.487; no difference
1
54. A lottery is conducted that involves the random selection of numbers from 0 to 4. To make sure that the lottery is fair,
a sample of 250 was taken. The following results were obtained:
Value
Frequency
0
40
1
45
2
55