CHAPTER 19: NONPARAMETRIC STATISTICS
TRUE/FALSE
1. The Wilcoxon rank sum test is used to compare two populations when the samples are independent
and the data are either ordinal or interval but not normally distributed.
2. The procedure for the Wilcoxon rank sum test requires that we rank each group separately rather than
together.
3. A parametric test is a hypothesis test that depends on certain specific assumptions about the probability
distribution of population values or the sizes of population parameters.
4. Nonparametric tests are methods of inference that make no assumptions whatsoever about the nature
of underlying population distributions or their parameters.
5. Statistical tests that are not very sensitive to errors in assumptions are called parametric tests.
6. In a normal approximation to the Wilcoxon rank sum test, the standardized test statistic is calculated as
z = 1.96. For a two-tail test, the p-value of the test is 0.025.
7. The z-test approximation to the Wilcoxon rank sum test for two independent samples requires that both
sample sizes are larger than 10.
8. Statistical methods that require few assumptions, if any, about the population distribution are known as
parametric techniques.
9. 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 independent.
10. Nonparametric methods are designed to test ordinal data.
MULTIPLE CHOICE
1. 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.
larger than 30.
c.
drawn from normal populations.
b.
independent.
d.
drawn from matched pairs experiment.
2. The Wilcoxon rank sum test statistic T is approximately normally distributed whenever the sample
sizes are larger than:
a.
25
c.
15
b.
20
d.
10
3. 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.
c.
largest observation.
b.
middle observation.
d.
observation that occurs most frequently.
4. Statistical methods that require few assumptions, if any, about the population distribution are
known as:
a.
parametric techniques.
c.
free agent techniques.
b.
nonparametric techniques.
d.
All of these choices are true.
5. The nonparametric tests discussed in your book (Wilcoxon rank sum test, sign test, Wilcoxon signed
rank sum test, Kruskal-Wallis test, and Friedman test) all require that the probability distributions be:
a.
identical except with respect to shape (distribution).
b.
identical except with respect to spread (variance).
c.
identical except with respect to location.
d.
different with respect to location, spread, and shape.
6. In a Wilcoxon rank sum test, the two sample sizes are 4 and 6, and the value of the Wilcoxon test
statistic is T = 20. If the test is a two-tail and the level of significance is
= 0.05, 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.
7. In a normal approximation to the Wilcoxon rank sum test, the standardized test statistic is calculated as
z = 1.80. For a two-tail test, the p-value is:
a.
0.0359
c.
0.2321
b.
0.4641
d.
0.0718
8. Statistical methods that require, among other assumptions, that the populations be normally distributed
are known as:
a.
distribution-free techniques.
c.
parametric techniques.
b.
nonparametric techniques.
d.
both a and b are correct.
9. Which of the following will never be a required condition of a nonparametric test?
a.
Data are ordinal.
b.
Data are interval.
c.
The samples are drawn from normally distributed populations.
d.
The populations being compared are identical in spread and shape.
10. You are performing the Wilcoxon rank-sum test. The 13th through 15th values in an ordered array of
pooled sample data all equal $180 (while the 12th value is less and the 16th value is more). The
appropriate ranks for the three $180 values are:
a.
14, 14, 14
c.
12.5, 13, 14.5
b.
13, 14, 15
d.
12.5, 13, 15
11. A nonparametric method that is equivalent to the Wilcoxon rank sum test is the:
a.
Wilcoxon signed rank sum test.
c.
Kruskal-Wallis test.
b.
Friedman test.
d.
Mann-Whitney test.
12. Consider the following data set: 2.2, 2.3, 2.3, 2.5, 2.6, 2.7, 2.8, 2.8, 2.8, 2.9, 3.1, 3.2, and 3.5. The rank
assigned to the three observations of value 2.8 is:
a.
6
c.
8
b.
7
d.
9
13. Comparing the output of strawberries grown on plots using fertilizer A with that grown on otherwise
identical plots using fertilizer B, in order to make a general assessment of relative fertilizer
effectiveness, may well call for a:
a.
Friedman test.
c.
Wilcoxon rank-sum test.
b.
Kruskal-Wallis test.
d.
Spearman rank correlation test.
14. Consider the following data set: 14, 14, 15, 16, 18, 19, 19, 20, 21, 22, 23, 25, 25, 25, 25, and 28. The
rank assigned to the four observations of value 25 is:
a.
12
c.
13
b.
12.5
d.
13.5
COMPLETION
1. Nonparametric methods are designed to test ____________________ data.
2. Nonparametric procedures are often called ____________________-free statistics.
3. Instead of testing whether population means differ, a nonparametric technique would test whether the
population ____________________ differ.
4. The Wilcoxon rank sum test requires that the samples from the two populations be
____________________.
5. H0 in the Wilcoxon rank sum test is that the two population ____________________ are the same.
6. In the Wilcoxon rank sum test statistic, let T represent the total rankings of sample 1 amongst all
rankings. If T is small, this suggests that the location of population 1 is to the ____________________
of the location of population 2.
7. When the sample sizes from both populations are larger than ____________________, the Wilcoxon
rank sum statistic has an approximate normal distribution.
8. When the sample sizes from both populations are larger than 10, the Wilcoxon rank sum statistic has
an approximate ____________________ distribution.
SHORT ANSWER
1. In testing the hypotheses
H0: The two population locations are the same.
H1: The two population locations are different.
The critical values and sample sizes are n1 = 5, TL = 22, n2 = 9, and TU = 53.
a.
Which test is used for testing the hypotheses above?
b.
What is the
-level of this test?
The Wilcoxon rank sum test
b.
0.05
2. Because of the rising costs of construction accidents many construction firms have instituted safety
courses. Employees are encouraged to take these courses designed to heighten safety awareness. A
company is trying to decide which one of two courses to institute. To help make a decision, eight
employees take course 1 and another eight take course 2. Each employee writes a test, which is
marked, out of a possible 25. The results are shown below. Does this 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.
Course 1
Course 2
14
20
21
18
17
22
14
15
17
23
19
21
20
19
16
15
3. In testing 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.
The statistics nA = 6, TA = 32, nB = 8, and TB = 58 are calculated with data drawn from two independent
samples:
a.
Which test is used for testing the hypotheses above?
b.
What is the p-value of this test?
The Wilcoxon rank sum test
b.
4. 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 CEO 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 CEO to conclude at the 5% significance level that her belief is correct?
This Year
Last Year
5
5
6
5
4
3
5
3
5
4
4
3
5. 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
60
67
54
69
Sample B:
80
84
100
74
90
6. One of the required conditions of the sign test is that the number of nonzero differences n must be
greater than or equal to 30.
7. The sign test is always to be preferred to the t-test.
8. The sign test is employed to compare two populations when the experimental design is matched pairs,
and the data are ordinal but not normally distributed.
9. A two-sample t–test is the parametric counterpart of the Wilcoxon signed rank sum test for matched
pairs.
10. A two independent sample t-test corresponds to a Wilcoxon signed rank sum test for paired samples.
11. The Wilcoxon signed rank sum test for matched pairs is the nonparametric counterpart of the paired
two-sample t-test of
D.
12. The Wilcoxon signed rank sum test is applied to compare two populations, when the samples are
matched pairs and the data are interval but not normally distributed.
13. The normal approximation to the Wilcoxon signed rank sum test statistic is used whenever the sample
size is large (more than 30).
14. Some statistical software perform the Mann-Whitney test. However, this test and the sign test are
equivalent tests.
15. Which of the following correctly describes the sign test?
a.
It often uses the directions of differences observed in matched-pairs sample to determine
whether the relative frequency distributions of two statistical populations are identical to
or different from one another.
b.
It is often used to determine whether a sample comes from a population with a specified
median.
c.
Both a and b.
d.
Neither a nor b.
16. The sign test statistic has what kind of distribution?
a.
normal
b.
binomial
c.
nonparametric
d.
None of these choices.
17. A nonparametric test for comparing two or more populations should be used instead of its parametric
counterpart if:
a.
no information about the shape of the populations is available.
b.
the sample sizes are large.
c.
the populations have equal means.
d.
the populations are normally distributed.
18. The sign test statistic is approximately normally distributed whenever the sample sizes are larger than:
a.
30
b.
10
c.
5
d.
None of these choices.
19. A nonparametric method to compare two populations, when the samples are matched pairs and the
data are ordinal, is the:
a.
sign test.
b.
matched pairs t–test
c.
Wilcoxon rank sum test.
d.
Wilcoxon signed rank sum test..
20. The nonparametric counterpart of the parametric t-test of
D for matched pairs is the:
a.
Friedman test.
b.
Kruskal-Wallis test.
c.
Wilcoxon signed rank sum test.
d.
Wilcoxon rank sum test.
21. A nonparametric method to compare two populations, when the samples are matched pairs and the
data are interval, and 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.
22. The Wilcoxon signed rank sum test statistic is approximately normally distributed whenever the
sample sizes are larger than:
a.
5
b.
10
c.
30
d.
None of these choices.
ANS:
C
23. 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.
reject the null hypothesis.
b.
don’t reject the null hypothesis.
c.
the test results are inconclusive.
d.
perform a parametric test.
24. When the data are interval, how does the Wilcoxon signed rank sum test compare to the sign test?
a.
Wilcoxon uses the information in the data more effectively.
b.
Wilcoxon uses the information in the data less effectively.
c.
Cannot compare the two tests without knowing the sample size.
d.
None of these choices.
25. Which of the following tests employs matched-pairs sampling?
a.
The Mann-Whitney test.
b.
The Wilcoxon rank-sum test.
c.
The Wilcoxon signed-rank sum test.
d.
None of these choices.
26. When data are ordinal and you want to analyze matched pairs differences, you use the
____________________ test.
27. When data are interval, but not normal, and you want to analyze matched pairs differences, use the
____________________ test.
28. The Wilcoxon signed rank sum test incorporates not only the sign of each difference, but the
____________________ of the difference as well.
29. The sign test statistic has a(n) ____________________ distribution.
30. When n is greater than or equal to 10, the sign test statistic has an approximate
____________________ distribution.
31. For the sign test, n equals the number of differences in the sample that are ____________________.
32. The Wilcoxon signed rank sum test requires you to rank the absolute ____________________.
Campaign Commercials
It is important to sponsors of television shows that viewers remember as much as possible about the
commercials. The campaign manager for a local politician is trying to decide which of two
commercials to use on 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
33. {Campaign Commercials Narrative} Which test is appropriate for this situation?
34. {Campaign Commercials Narrative} Does this data provide enough evidence at the 5% significance
level to conclude that the responses to the two commercials differ?
Keyboard Speed
Ten administration assistants were selected at random from a large university. The keyboard speed
(number of words per minute) was recorded for each secretary on two different brands of computer
keyboards. Assume that the typing speeds are not normally distributed. The following results were
obtained.
Secretary
Brand A
Brand B
Barry
72
74
Betty
80
86
Carol
68
72
Cliff
74
70
Ellen
86
85
Fred
75
73
Gwen
78
72
Harry
69
65
Ingrid
76
79
Jerome
65
64
35. {Keyboard Speed Narrative} Perform the sign test to determine if these data provide enough evidence
at the 5% significance level to infer that the brands differ with respect to keyboard speed.
36. {Keyboard Speed Narrative} Perform the Wilcoxon signed rank sum test at the 5% significance level.
Movie Scripts
37. {Movie Scripts Narrative} Which test is appropriate for this situation?
38. {Movie Scripts Narrative} Can Big-Screen conclude at the 5% significance level that script 2 is more
highly rated than script 1?
39. {Movie Scripts Narrative} What is the p-value of this test?
40. {Movie Scripts Narrative} Explain how to use the p-value for testing the hypotheses at the 5%
significance level.
Coffee
41. {Coffee Narrative} Which test is appropriate for this situation?
42. {Coffee Narrative} Can we conclude at the 1% significance level that the general managers’ claim is
false?
43. {Coffee Narrative} What is the p-value of the test in the previous question?
44. The Kruskal-Wallis test can be conducted as one or two-tail test.
45. A one-sample t-test is the parametric counterpart of the Kruskal-Wallis test.
46. The Kruskal-Wallis test is an extension of the Wilcoxon rank-sum test from two to more than two
statistical populations.
47. The Kruskal-Wallis test is applied to compare two or more populations, when the samples are
independent and the data are either ordinal or interval but not normal.
48. The critical value is taken from the F-distribution whenever the test is a Kruskal-Wallis test.
49. Suppose using Kruskal-Wallis test, there are five samples, and the value of the test statistic is
calculated as H = 12.32. Then the most accurate statement that can be made about the p-value of the
test is that it is greater than 0.025 but smaller than 0.05
50. The Kruskal-Wallis test can be used to test for a difference between two populations. It will produce
the same outcome as the two-tail Wilcoxon rank sum test.
51. The Kruskal-Wallis test can be used to determine whether a difference exists between two populations.
However, to determine whether one population location is larger than another, we must apply the
Wilcoxon rank sum test.
52. We can use the Friedman test to determine whether a difference exists between two populations.
However, if we want to determine whether one population location is larger than another, we must use
the sign test.
53. The Friedman test is employed to compare two or more populations when the data are generated from
a randomized block experiment, and are either ordinal or interval, but not normally distributed.
54. A one-sample t-test is the parametric counterpart of the Friedman test for randomized block
experimental design.
55. The Friedman test is the nonparametric counterpart of the randomized block experimental design of
the analysis of variance.
56. If Friedman test is applied to a data set that are generated from a randomized block experiment with 3
treatments and 5 blocks, then the rejection region at the 2.5% significance level is Fr > 12.8325.
57. The Friedman test statistic is approximately chi-squared distributed with (k − 1) degrees of freedom,
provided that either the number of blocks b or the number of treatments k is greater than or equal to 5.
58. The Friedman test statistic is approximately chi-squared distributed with (k − 1) degrees of freedom,
provided that both the number of blocks b and the number of treatments k are greater than or equal to
5.
59. We can use the Friedman test to determine whether two populations differ. The conclusion will be the
same as that produced from the sign test.
60. The alternative hypothesis for the Friedman test is that all the population locations differ.
61. In all applications of the Kruskal-Wallis test, the alternative hypothesis to be tested is always stated as:
a.
the locations of all k populations are the same.
b.
the locations of all k populations differ.