CHAPTER 12—TESTS OF GOODNESS OF FIT, INDEPENDENCE AND
MULTIPLE PROPORTIONS
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 sampling distribution for a goodness of fit test is
a.
the Poisson distribution
b.
the t distribution
c.
the normal distribution
d.
the chi-square distribution
3. 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.
4. 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.
5. 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
6. 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
7. 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.
8. The degrees of freedom for a contingency table with 12 rows and 12 columns is
a.
144
b.
121
c.
12
d.
120
9. The degrees of freedom for a contingency table with 6 rows and 3 columns is
a.
18
b.
15
c.
6
d.
10
Exhibit 12-1
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.
10. Refer to Exhibit 12-1. The expected frequency for each group is
a.
0.333
b.
0.50
c.
1/3
d.
50
11. Refer to Exhibit 12-1. The calculated value for the test statistic equals
a.
2
b.
-2
c.
20
d.
4
12. Refer to Exhibit 12-1. The number of degrees of freedom associated with this problem is
a.
150
b.
149
c.
2
d.
3
13. Refer to Exhibit 12-1. 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
14. Refer to Exhibit 12-1. 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.
Exhibit 12-2
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.
15. Refer to Exhibit 12-2. The expected number of freshmen is
a.
83
b.
90
c.
30
d.
10
16. Refer to Exhibit 12-2. The expected frequency of seniors is
a.
60
b.
20%
c.
68
d.
64
17. Refer to Exhibit 12-2. The calculated value for the test statistic equals
a.
0.5444
b.
300
c.
1.6615
d.
6.6615
18. Refer to Exhibit 12-2. 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
19. Refer to Exhibit 12-2. The null hypothesis
a.
should not be rejected
b.
should be rejected
c.
was designed wrong
d.
None of these alternatives is correct.
Exhibit 12-3
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
70
10
20
50
We are interested in determining whether or not the medication was effective in curing the common
cold.
20. Refer to Exhibit 12-3. The expected frequency of those who received medication and were cured is
a.
70
b.
150
c.
28
d.
48
21. Refer to Exhibit 12-3. The test statistic is
a.
10.08
b.
54.02
c.
1.96
d.
1.645
22. Refer to Exhibit 12-3. The number of degrees of freedom associated with this problem is
a.
4
b.
149
c.
1
d.
3
23. Refer to Exhibit 12-3. 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
24. Refer to Exhibit 12-3. The null hypothesis
a.
should not be rejected
b.
should be rejected
c.
should be revised
d.
None of these alternatives is correct.
25. The degrees of freedom for a contingency table with 10 rows and 11 columns is
a.
100
b.
110
c.
21
d.
90
Exhibit 12-4
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.
26. Refer to Exhibit 12-4. This problem is an example of a
a.
normally distributed variable
b.
test for independence
c.
Poisson distributed variable
d.
multinomial population
27. Refer to Exhibit 12-4. The expected frequency for the Business College is
a.
0.3
b.
0.35
c.
90
d.
105
28. Refer to Exhibit 12-4. The calculated value for the test statistic equals
a.
0.01
b.
0.75
c.
4.29
d.
4.38
29. Refer to Exhibit 12-4. 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
30. Refer to Exhibit 12-4. 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 12-5
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
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.
31. Refer to Exhibit 12-5. 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
32. Refer to Exhibit 12-5. The expected number of adults who prefer coffee is
a.
0.25
b.
0.33
c.
150
d.
200
33. Refer to Exhibit 12-5. The calculated value for this test for independence is
a.
0
b.
8.4
c.
62.5
d.
82.5
34. Refer to Exhibit 12-5. 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.
35. Excel’s ____ function is used to perform a goodness of fit test.
a.
z-Test: Two Sample for Means
b.
t-Test: Two Sample Assuming Equal Variances
c.
CHISQ.DIST.RT
d.
NORM.S.DIST
36. In a goodness of fit test, Excel’s CHISQ.DIST.RT function returns a
a.
chi-square critical value
b.
chi-square test statistic
c.
p-value
d.
confidence interval estimate
37. 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
38. Excel’s CHISQ.DIST function can be used to perform
a.
a multinomial goodness of fit test
b.
a Poisson distribution goodness of fit test
c.
a Normal distribution goodness of fit test
d.
All of the other ANSWERs are correct
Exhibit 12-6
The following shows the number of individuals in a sample of 300 who indicated they support the new
tax proposal.
Political Party
Support
Democrats
100
Republicans
120
Independents
80
We are interested in determining whether or not the opinions of the individuals of the three groups are
uniformly distributed.
39. Refer to Exhibit 12-6. The expected frequency for each group is
a.
0.333
b.
0.50
c.
50
d.
None of these alternatives is correct.
40. Refer to Exhibit 12-6. The calculated value for the test statistic equals
a.
300
b.
4
c.
0
d.
8
41. Refer to Exhibit 12-6. The number of degrees of freedom associated with this problem is
a.
2
b.
3
c.
300
d.
299
42. The test statistic for goodness of fit has a chi-square distribution with k – 1 degrees of freedom
provided that the expected frequencies for all categories are
a.
5 or more
b.
10 or more
c.
k or more
d.
2k
43. The test for goodness of fit
a.
is always a lower-tail test
b.
is always an upper-tail test
c.
is always a two-tailed test
d.
can be a lower- or upper-tail test
44. The number of categories of outcomes per trial for a multinomial probability distribution is
a.
two or more
b.
three or more
c.
four or more
d.
five or more
45. The test for goodness of fit, test of independence, and test of multiple proportions are designed for use
with
a.
categorical data
b.
bivariate data
c.
quantitative data
d.
ordinal data
46. 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
PROBLEM
1. 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%
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.
2. 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 = 0.05.
3. 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.
4. 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 = 0.05.
5. In 2012, 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 2013, 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 2012
and 2013.
6. 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 = .01 to see if the proportion of voters who favored the various candidates
changed.
7. 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 = 0.05,
test to see if the methods of payment have changed during the Christmas rush.
8. 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 = 0.05, test the manufacturer’s claim.
9. 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
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?
10. 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?
a.
9.21034
d.
Do not reject the null hypothesis, there is not enough evidence to conclude that the market
share has changed
11. 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
a.
b.
5
9.48773
d.
Do not reject the null hypothesis, there is not enough evidence to conclude that the lottery is
unfair