Chapter 16— Nonparametric techniques: Comparing two populations
MULTIPLE CHOICE
1. Statistical methods that require, among other assumptions, that the populations be normally distributed
are known as:
A.
distribution-free statistics.
B.
nonparametric statistics.
C.
parametric statistics.
D.
Both A and B are correct answers.
2. A nonparametric method to compare two populations, when the samples are independent and the data
are either ranked or quantitative where the normality requirement necessary to perform the parametric
test is not satisfied, is the:
A.
Wilcoxon rank sum test.
B.
sign test.
C.
Wilcoxon signed rank sum test.
D.
equal-variance t–test of μ1 – μ2.
3. A nonparametric method to compare two populations, when the samples are matched pairs and the
data are quantitative where the normality requirement necessary to perform the parametric test is
unsatisfied, is the:
A.
Wilcoxon rank sum test.
B.
sign test.
C.
matched pairs t–test.
D.
Wilcoxon signed rank sum test.
4. A nonparametric method to compare two populations, when the samples are matched pairs and the
data are ranked, is the:
A.
Wilcoxon signed rank sum test.
B.
sign test.
C.
Wilcoxon rank sum test.
D.
matched pairs t–test.
5. You can use a nonparametric procedure when:
A.
the assumptions required by the parametric procedures are not met.
B.
the data represent ordinal measurements.
C.
the hypotheses do not involve a parameter.
D.
All of the above are correct.
6. The Wilcoxon rank sum test (like most of the nonparametric tests presented in your textbook) actually
tests to determine whether the population distributions have identical:
A.
locations.
B.
spreads (variances).
C.
shapes.
D.
All of the above are correct answers.
7. The first step in a Wilcoxon rank sum test is to combine the data values in the two samples and assign
a rank of 1 to the:
A.
smallest observation.
B.
middle observation.
C.
largest observation.
D.
observation that occurs most frequently.
8. Nonparametric procedures are often called:
A.
distribution-free methods.
B.
inferential statistics.
C.
chi-squared tests.
D.
None of the above answers is correct.
9. Which of the following will never be a required condition of a nonparametric test?
A.
The data are ranked.
B.
The data are quantitative.
C.
The samples are drawn from normally distributed populations.
D.
The populations being compared are identical in spread and shape.
10. In a Wilcoxon rank sum test, the two sample sizes are 4 and 6, and the value of the Wilcoxon test
statistic is T = 10. If the test is two-tailed and the level of significance is = 0.5, then:
A.
the null hypothesis will be rejected.
B.
the null hypothesis will not be rejected.
C.
the alternative hypothesis will not be rejected.
D.
Not enough information has been given to answer this question.
11. To apply the Wilcoxon rank sum test to determine whether the location of population 1 is different
from the location of population 2, the samples must be:
A.
from normal populations.
B.
from a matched pairs experiment.
C.
independent.
D.
larger than 30.
12. In a normal approximation to the Wilcoxon rank sum test, the standardised test statistic is calculated as
z = 1.80. For a two-tail test, the p-value is:
A.
0.0359.
B.
0.4641.
C.
0.2321.
D.
0.0718.
13. In the sign test applications, the normal approximation to the binomial distribution may be used
whenever the number of nonzero differences is greater than or equal to:
A.
30.
B.
20.
C.
10.
D.
5.
14. In a normal approximation to the sign test, the standardised test statistic is calculated as: z = –1.58. To
test the alternative hypothesis that the location of population 1 is to left of the location of population 2,
the p-value is:
A.
0.1142.
B.
0.2215.
C.
0.0571.
D.
0.2284.
15. The Wilcoxon rank sum test statistic T is approximately normally distributed whenever the sample
sizes are larger than:
A.
10.
B.
15.
C.
20.
D.
30.
16. In a normal approximation to the Wilcoxon signed rank test, the test statistic is calculated as
z = 2.36. For a two-tail test, the p-value is:
A.
0.0366.
B.
0.0183.
C.
0.4817.
D.
0.0092.
17. In a Wilcoxon signed rank sum test, the test statistic is calculated as T = 75. If there are n = 15
observations for which D 0, and a two-tail test is performed at the 5% significance level, then:
A.
we reject the null hypothesis.
B.
we don’t reject the null hypothesis.
C.
the test results are inconclusive.
D.
we perform a parametric test.
18. In a Wilcoxon signed rank sum test, the test statistic is calculated as T = 45. The alternative hypothesis
is stated as: The location of population 1 is to the right of the location of population 2. If there are n =
10 observations for which D 0, and the 5% significance level is used, then the:
A.
null hypothesis will be rejected.
B.
null hypothesis will not be rejected.
C.
test results are inconclusive.
D.
t-test for matched pairs must be used.
19. In a sign test, the following information is given: number of zero differences = 3, number of positive
differences = 20, and number of negative differences = 5. The value of the standardised test statistic is:
A.
5.
B.
4.
C.
3.
D.
2.
20. The significance level for a Wilcoxon signed rank sum test is 0.05. The alternative hypothesis is stated
as: The location of population 1 is to the left of the location of population 2. The appropriate critical
value for a sample of size 20 (that is, the number of nonzero differences) is:
A.
158.
B.
150.
C.
60.
D.
52.
21. Consider the following data set:
11
11
12
13
15
16
16
17
18
19
20
21
22
23
23
25
The rank assigned to the two observations of value 16 is:
A.
5.5.
B.
6.
C.
6.5.
D.
7.
22. Consider the following data set:
1.2
1.3
1.3
1.5
1.6
1.7
1.8
1.8
1.8
1.9
2.1
2.2
2.5
The rank assigned to the three observations of value 1.8 is:
A.
6.
B.
7.
C.
8.
D.
9.
23. Consider the following two samples:
A:
14
B:
13
17
The value of the test statistic for a right-tail Wilcoxon rank sum test is:
A.
3.
B.
7.
C.
11.
D.
44.
24. Consider the following two samples:
A:
14
15
17
20
45
B:
25
29
32
35
38
The value of the test statistic for a left-tail Wilcoxon rank sum test is:
A.
6.
B.
20.
C.
35.
D.
55.
25. A matched pairs experiment yielded the following paired differences:
0
–2
2
1
–3
2
0
3
1
–1
–2
3
1
–1
1
2
3
0
The value of the standardised sign test statistic is:
A.
0.4714.
B.
1.8856.
C.
0.8856.
D.
1.2910.
26. The significance level for a Wilcoxon signed rank sum test is 0.05. The alternative hypothesis is stated
as: The location of population 1 is to the right of the location of population 2. The appropriate critical
value for a sample of size 20 (that is, the number of nonzero differences is 20) is:
A.
158.
B.
150.
C.
60.
D.
52.
27. A Wilcoxon rank sum test for comparing two populations involves two independent samples of sizes 5
and 7. The alternative hypothesis is stated as: The location of population 1 is different from the
location of population 2. The appropriate critical values at the 5% significance level are:
A.
20, 45.
B.
22, 43.
C.
33, 58.
D.
35, 56.
28. In a Wilcoxon signed rank sum test for matched pairs with n = 35, the rank sums of the positive and
negative differences are 380 and 225, respectively. The value of the standardised test statistic is:
A.
1.689.
B.
1.065.
C.
1.206.
D.
1.400.
29. A Wilcoxon rank sum test for comparing two populations involves two independent samples of sizes
15 and 20. The unstandardised test statistic (that is the rank sum) is T = 210. The value of the
standardised test statistic is:
A.
14.
B.
10.5.
C.
6.
D.
–2.
TRUE/FALSE
1. The Wilcoxon rank sum test for independent samples is used to replace the equal-variances test of
1 2
 
−
when the sample sizes
1
n
and
2
n
are small, but equal.
2. The Wilcoxon rank sum test for independent samples is used to replace the equal-variances test of
1 2
 
−
when the populations are not normally distributed.
3. The Wilcoxon signed rank sum test for matched pairs is used to replace the t-test of
D
if the
differences are very non-normal.
SHORT ANSWER
1. Use the Wilcoxon rank sum test on the data below to determine at the 5% significance level whether
the location of population A is to the left of the location of population B.
Sample A:
75
69
Sample B:
70
82
2. Given the following statistics, use the Wilcoxon rank sum test to determine at the 5% significance
level whether the location of population A is to the right of the location of population B.
TA = 42, nA = 6, TB = 36, nB = 9.
3. Use the Wilcoxon rank sum test on the data below to determine at the 10% significance level whether
the two population locations differ.
Sample 1:
32
22
19
29
20
34
25
9
28
17
Sample 2:
29
20
18
27
19
23
19
12
22
10
4. Use the Wilcoxon rank sum test on the data below to determine at the 10% significance level whether
the two population locations differ.
Sample 1:
17
20
16
Sample 2:
17
25
15
5. Use the 5% significance level to test the hypotheses:
H0: The two population locations are the same
H1: The location of population A is to the left of the location of population B,
given that the data below are drawn from two independent samples.
Sample A:
9
11
9
10
12
8
Sample B:
8
7
5
7
9
5
6
6. Use the 5% significance level to test the hypotheses:
H0: The two population locations are the same
H1: The location of population A is to the left of the location of population B,
given that the data below are drawn from a matched pairs experiment:
Matched pair
1
2
3
4
5
6
7
8
A
5
11
10
8
9
10
3
10
B
6
9
12
10
12
10
5
8
7. The following data represent the test scores of eight students in a statistics test before and after
attending extra help sessions for the test.
Student
Before
After
Andrew
82
90
Brenda
75
86
Carmen
90
90
David
68
62
Edward
87
89
Frank
73
75
Gill
81
78
Heather
92
98
Use the Wilcoxon signed rank sum test to determine at the 5% significance level whether the extra
help sessions have been effective.
8. The following statistics are drawn from two independent samples:
TA = 800, nA = 25, TB = 1100, nB = 28.
Test at the 5% significance level to determine whether the two population locations differ.
9. Perform the Wilcoxon signed rank sum test for the following matched pairs to determine at the 10%
significance level whether the two population locations differ.
Matched
pair
1
2
3
4
5
6
7
A
13
9
11
10
12
8
14
B
11
10
10
6
10
4
12
10. Use the 5% significance level to test the hypotheses:
H0: The two population locations are the same
H1: The two population locations are different,
given that the data below are drawn from a matched pairs experiment.
Matched
pair
1
2
3
4
5
6
7
8
9
10
A
32
15
19
25
39
18
26
41
33
23
B
28
14
20
20
27
23
25
31
25
23
11. In general, before an academic publisher agrees to publish a book, each manuscript is thoroughly
reviewed by university lecturers. Suppose that the Cengage Australia company has recently received
two manuscripts for statistics books. To help them decide which one to publish, both are sent to 30
lecturers of statistics, who rate the manuscripts to judge which one is better. Suppose that 12 lecturers
rate manuscript 1 better, i.e. assign a higher score to manuscript 1 than to manuscript 2, and 18 rate
manuscript 2 better.
a. Which test is appropriate for this situation?
b. Can Cengage Australia conclude at the 5% significance level that manuscript 2 is more highly
rated than manuscript 1?
c. What is the p-value of this test?
12. A supermarket chain has its own home brand of ice cream. The general manager claims that her ice
cream is better than the ice cream sold by a well-known ice cream company. To test the claim, 40
individuals are randomly selected to participate in the following experiment. Each respondent is given
the two brands of ice cream to taste (without any identification) and asked to judge which one is better.
Suppose that 25 people judge the well-known ice cream brand to be better, 4 say that the brands taste
the same, and the rest claim that the supermarket brand is better.
a. Which test is appropriate for this situation?
b. Can we conclude at the 1% significance level that the general manager’s claim is false?
c. What is the p-value of this test?
13. In testing the hypotheses:
H0: The two population locations are the same
H1: The location of population A is to the right of the location of population B,
with data drawn from two independent samples, the following statistics are calculated:
nA = 5, TA = 49, nB = 8, TB = 42.
a. Which test is used for testing the hypotheses above?
b. What is the p-value of this test?
14. In testing the hypotheses:
H0: The two population locations are the same
H1: The two population locations are different,
with data drawn from two independent samples, the following statistics are calculated:
nA = 5, TA = 22, nB = 9, TB = 83.
a. Which test is used for testing the hypotheses above?
b. What is the p-value of this test?
15. In testing the hypotheses:
H0: The two population locations are the same
H1: The two population locations are different,
with data drawn from a matched pairs experiment, the following statistics are calculated:
n = 40, T+ = 238, T– = 582.
a. Which test is used for testing the hypotheses above?
b. What is the p-value of this test?
16. In testing the hypotheses:
H0: The two population locations are the same
H1: The location of population A is to the right of the location of population B,
with data drawn from a matched pairs experiment, the following statistics are calculated:
n = 23, T+ = 193, T– = 83.
a. Which test is used in testing the hypotheses above?
b. What is the p-value of this test?
17. In recent years, airlines have been subjected to various forms of criticism. An executive of Airline X
has taken a quick poll of 16 regular airline passengers. Each passenger was asked to rate the airline he
or she last flew on. The ratings were on a 7-point Likert scale, where 1 = poor and 7 = very good. Of
the 16 respondents, six last flew on Airline X and the remainder flew on other airlines. The ratings are
shown below. Can the executive conclude from these data, with 5% significance, that Airline X is
more highly rated than the other airlines?
Ratings of airlines
Airline X
Other airlines
6
5
4
3
5
3
6
2
5
3
3
4
3
5
3
1
18. Because of the rising costs of industrial accidents, many chemical, mining and manufacturing firms
have instituted safety courses. Employees are encouraged to take these courses, which are designed to
heighten safety awareness. A company is trying to decide which of two courses to institute. To help
make a decision, eight employees take course 1 and another eight take course 2. Each employee sits a
test, which is marked out of a possible 25. The results are shown below. Do these data provide
sufficient evidence at the 5% level of significance to conclude that the marks from course 2 are higher
than those of course 1? Assume that the scores are not normally distributed.
Safety test scores
Course 1
Course 2
14
20
21
18
17
22
14
15
17
23
19
21
20
19
16
15
19. Each year, the Human Resources department in a large corporation assesses the performance of all of
its employees. Each employee is rated for various aspects of his or her job on a 7-point scale, where
1 = very unsatisfactory and 7 = satisfactory. The president of the company believes that the assessment
scores this year are lower than last year’s. To examine the validity of this belief, she draws a random
sample of six employees’ scores from last year and another six employees’ scores this year. Do the
data listed below allow the president to conclude at the 5% significance level that her belief is correct?
Employees ratings scores
This year
Last year
6
5
5
5
4
3
5
3
5
4
4
3
20. A matched pairs experiment yielded the following results:
Number of positive differences = 20.
Number of zero differences = 2.
Number of negative differences = 8.
Can we infer at the 5% significance level that the location of population 1 is to the right of the location
of population 2?
21. It is important to sponsors of television shows that viewers remember as much as possible about the
commercials. The advertising executive of a large company is trying to decide which of two
commercials to use in a weekly half-hour comedy. To help make a decision, she decides to have 12
individuals watch both commercials. After each viewing, each respondent is given a quiz consisting of
10 questions. The number of correct responses is recorded and listed below. Assume that the quiz
results are not normally distributed.
Quiz scores
Respondent
Commercial 1
Commercial 2
1
7
9
2
8
9
3
6
6
4
10
10
5
5
4
6
7
9
7
5
7
8
4
5
9
6
8
10
7
9
11
5
6
12
8
10
a. Which test is appropriate for this situation?
b. Do these data provide enough evidence at the 5% significance level to conclude that the two
commercials differ?
22. Ten secretaries were selected at random from among the secretaries of a large university. The typing
speed (number of words per minute) was recorded for each secretary on two different brands of
computer keyboards. The following results were obtained.
Computer Keyboard
Secretary
Brand A
Brand B
Amy
71
84
Betty
63
74
Carol
78
71
Donna
87
87
Ellen
87
71
Faith
89
86
Gwen
60
85
Heather
72
87
Ingrid
86
74
Jody
64
85
Assume that the typing speeds are not normally distributed. Perform the sign test to determine whether
these data provide enough evidence at the 5% significance level to infer that the brands differ with
respect to typing speed.
23. Ten secretaries were selected at random from among the secretaries of a large university. The typing
speed (number of words per minute) was recorded for each secretary on two different brands of
computer keyboards. The following results were obtained.
Computer Keyboard
Secretary
Brand A
Brand B
Amy
71
84
Betty
63
74
Carol
78
71
Donna
87
87
Ellen
87
71
Faith
89
86
Gwen
60
85
Heather
72
87
Ingrid
86
74
Jody
64
85
Perform the Wilcoxon signed rank sum test at the 5% level of significance.
24. If in a matched pairs experiment, there are 25 negative, 5 zero, and 16 positive differences. Perform
the sign test at the 5% significance level to determine whether the two population locations differ.