CHAPTER 18—NONPARAMETRIC METHODS
MULTIPLE CHOICE
1. Statistical methods that generally require very few, if any, assumptions about the population
distribution are known as
a.
parametric
b.
nonparametric
c.
free methods
d.
None of these alternatives is correct.
2. Which of the following tests would not be an example of nonparametric method?
a.
Mann-Whitney-Wilcoxon test
b.
Wilcoxon signed-rank test
c.
sign test
d.
t test
3. A nonparametric version of the Parametric analysis of variance test is the
a.
Kruskal-Wallis Test
b.
Mann-Whitney-Wilcoxon Test
c.
sign test
d.
Wilcoxon signed-rank test
4. A nonparametric method for determining the differences between two populations based on two
matched samples where only preference data is required is the
a.
Mann-Whitney-Wilcoxon test
b.
Wilcoxon signed-rank test
c.
sign test
d.
Kruskal-Wallis Test
5. When ranking combined data in a Wilcoxon signed rank test, the data that receives a rank of 1 is the
a.
lowest value
b.
highest value
c.
middle value
d.
average of the highest and the lowest of values
6. Statistical methods that require assumptions about the population are known as
a.
distribution free
b.
nonparametric
c.
either distribution free of nonparametric
d.
parametric
7. The Spearman rank-correlation coefficient is
a.
a correlation measure based on the average of data items
b.
a correlation measure based on rank-ordered data for two variables
c.
a correlation measure based on the median of data items
d.
None of these alternatives is correct.
8. The level of measurement that allows for the rank ordering of data items is
a.
nominal measurement
b.
ratio measurement
c.
interval measurement
d.
ordinal measurement
9. The level of measurement that is a label for the purpose of identifying an item is
a.
ordinal measurement
b.
ratio measurement
c.
nominal measurement
d.
internal measurement
10. The labeling of parts as “defective” or “non-defective” is an example of
a.
ordinal data
b.
ratio data
c.
interval data
d.
nominal data
11. A nonparametric test for the equivalence of two populations would be used instead of a parametric test
for the equivalence of the population parameters if
a.
the samples are very large
b.
the samples are not independent
c.
no information about the populations is available
d.
The parametric test is always used in this situation.
12. A nonparametric test would be used if
a.
nominal data is available
b.
interval data is available
c.
it is known that the population is normally distributed
d.
None of these alternatives is correct.
Exhibit 18-1
Ten people were given two types of cereal, Brand X and Brand Y. Three people preferred Brand X, 5
people preferred Brand Y, and 2 people were undecided. We want to determine whether or not the two
products are equal.
13. Refer to Exhibit 18-1. The null hypothesis that is being tested is
a.
H0: = 5
b.
H0: = 0.5
c.
H0: p = 5
d.
H0: p = 0.5
14. Refer to Exhibit 18-1. To test the null hypothesis, the appropriate probability distribution to use is
a.
normal
b.
chi-square
c.
Poisson
d.
binomial
15. Refer to Exhibit 18-1. The hypothesis is to be tested at the 5% level. The decision rule is not to reject
the null hypothesis if
a.
-1.96 < A < 1.96
b.
-2.262 < t < 2.262
c.
the number of “+” signs is greater than or equal to 2 and less than or equal to 6
d.
the number of “+” signs is greater than or equal to 1 and less than or equal to 7
16. Refer to Exhibit 18-1. The null hypothesis should be
a.
rejected
b.
not rejected
c.
revised
d.
None of these alternatives is correct.
Exhibit 18-2
Students in statistics classes were asked whether they preferred a 10-minute break or to get out of class
10 minutes early. In a sample of 150 students, 40 preferred a 10-minute break, 80 preferred to get out
10 minutes early, and 30 had no preference. We want to determine if there is a difference in students’
preferences.
17. Refer to Exhibit 18-2. To test the null hypothesis, the appropriate probability distribution to use is the
a.
normal
b.
chi-square
c.
t distribution
d.
binomial
18. Refer to Exhibit 18-2. The hypothesis is to be tested at the 5% level of significance. The decision rule
is not to reject the null hypothesis if
a.
-1.96 < z < 1.96
b.
-1.645 < z < 1.645
c.
z > 1.96 or z < -1.96
d.
the number of “+” signs is greater than or equal to 20 and less than or equal to 130
19. Refer to Exhibit 18-2. The test statistic based on the number of students who preferred to get out early
equals
a.
-3.65
b.
0.67
c.
0.82
d.
3.65
20. Refer to Exhibit 18-2. The null hypothesis should be
a.
rejected
b.
not rejected
c.
revised
d.
None of these alternatives is correct.
21. If a null hypothesis that states that two populations are identical is rejected using a nonparametric test,
then it is safe to assume that
a.
neither the means nor the variances are equal
b.
the means of the populations are not the same
c.
the variances of the populations are not the same
d.
We cannot be sure of the way in which the populations differ from each other.
Exhibit 18-3
A company advertises that food preparation time can be significantly reduced with the Handy Dandy
Slicer. A sample of 12 individuals prepared the ingredients for a meal with and without the slicer. You
are given the preparation times below.
Preparation Times
Person
With Slicer
Without Slicer
1
20
22
2
12
18
3
20
18
4
14
22
5
19
19
6
20
21
7
19
18
8
15
12
9
22
18
10
19
25
11
21
26
12
23
20
22. Refer to Exhibit 18-3. To test the null hypothesis, the appropriate probability distribution to use is
a.
normal
b.
chi-square
c.
t distribution
d.
binomial
23. Refer to Exhibit 18-3. The hypothesis is to be tested at the 5% level of significance. The decision rule
is not to reject the null hypothesis if
a.
-1.96 < z < 1.96
b.
-1.645 < z < 1.645
c.
z > 1.96 or z < -1.96
d.
the number of “+” signs is greater than or equal to 2 but less than or equal to 8
24. Refer to Exhibit 18-3. The test statistic equals
a.
-0.812 or 0.812
b.
-0.889 or 0.889
c.
-10 or 10
d.
-20 or 20
25. Refer to Exhibit 18-3. The null hypothesis should be
a.
rejected
b.
not rejected
c.
revised
d.
None of these alternatives is correct.
Exhibit 18-4
It has been hypothesized that there is no difference in the mathematical accuracy of men and women.
A sample of men and women were given math tests. The scores on the tests are given below.
Women
Men
Person
Score
Person
Score
1
95
1
80
2
86
2
87
3
100
3
93
4
100
4
95
5
99
5
97
6
98
6
82
7
88
7
89
8
92
8
86
9
94
9
75
10
89
10
82
11
79
26. Refer to Exhibit 18-4. To test the null hypothesis, the appropriate probability distribution to use is
a.
normal
b.
chi-square
c.
t distribution
d.
binomial
27. Refer to Exhibit 18-4. The null hypothesis is to be tested at the 5% level. The decision rule is not to
reject the null hypothesis if
a.
-1.96 < z < 1.96
b.
-1.645 < z < 1.645
c.
z > 1.96 or z < -1.96
d.
the number of “+” signs is greater than or equal to 2 but less than or equal to eight
28. Refer to Exhibit 18-4. The test statistic equals (using the women as population 1)
a.
-5.246
b.
0.176
c.
0.722
d.
2.5
29. Refer to Exhibit 18-4. Calculate a Spearman rank-correlation coefficient for 20 pairs of data when di2
= 50.
a.
0.0063
b.
0.0376
c.
0.9624
d.
0.9937
Exhibit 18-5
Forty-one individuals from a sample of 60 indicated they oppose legalized abortion. We are interested
in determining whether or not there is a significant difference between the proportions of opponents
and proponents of legalized abortion.
30. Refer to Exhibit 18-5. The null hypothesis that is being tested is
a.
H0: = 5
b.
H0: = 0.5
c.
H0: p = 5
d.
H0: p = 0.5
31. Refer to Exhibit 18-5. in this situation is
a.
60
b.
30
c.
41
d.
2
32. Refer to Exhibit 18-5. in this problem is
a.
15
b.
5.47
c.
3.87
d.
7.45
33. Refer to Exhibit 18-5. The test statistics is
a.
3.87
b.
2.84
c.
60
d.
0.5
34. Refer to Exhibit 18-5. The hypothesis is to be tested at the 5% level. The decision rule is not to reject
the null hypothesis if
a.
-1.96 < z < 1.96
b.
-3 < t < 3
c.
F > 5
d.
Chi-square > 5
35. Refer to Exhibit 18-5. The null hypothesis should be
a.
rejected
b.
not rejected
c.
Not enough information is given to answer this question.
d.
None of these alternatives.
36. Refer to Exhibit 18-5. The conclusion is that there
a.
is no significant difference between the proportions
b.
is a significant difference between the proportions
c.
could be a difference in proportions, depending on the sample size
d.
None of these alternatives is correct.
37. Excel’s ____ function can be used to conduct a sign test.
a.
POISSON.DIST
b.
T.INV
c.
BINOM.DIST
d.
CHISQ.DIST.RT
38. Excel’s ____ function can be used to conduct the Wilcoxon signed-rank test.
a.
POISSON.DIST
b.
BINOM.DIST
c.
CHISQ.DIST.RT
d.
NORM.S.DIST
39. Excel’s ____ function can be used to conduct the Kruskal-Wallis test.
a.
POISSON.DIST
b.
BINOM.DIST
c.
CHISQ.DIST.RT
d.
NORM.S.DIST
40. The Spearman rank-correlation coefficient for 20 pairs of data when di2 = 50 is.
a.
0.0063
b.
0.0376
c.
0.9624
d.
0.9937
Exhibit 18-6
It is believed that the median yearly income in a suburb of Atlanta is $70,000. A sample of 67
residents was taken. Thirty-eight had yearly incomes above $70,000, 26 had yearly incomes below
$70,000, and 3 had yearly incomes equal to $70,000. The null hypothesis to be tested is H0: median =
$70,000.
41. Refer to Exhibit 18-6. To test the null hypothesis, the appropriate probability distribution to use is
a.
normal
b.
chi-square
c.
t distribution
d.
binomial
42. Refer to Exhibit 18-6. The mean and the standard deviation (respectively) for this test about the
median are
a.
32 and 4
b.
32 and 16
c.
33.5 and 4
d.
33.5 and 16
43. Refer to Exhibit 18-6. The test statistic has a value of
a.
1.00
b.
1.50
c.
2.00
d.
2.50
44. Refer to Exhibit 18-6. The p-value for this test is
a.
0.4332
b.
0.8664
c.
0.0668
d.
0.1336
45. Refer to Exhibit 18-6. The null hypothesis should be
a.
rejected
b.
not rejected
c.
revised
d.
None of these alternatives is correct.
46. Nonparametric methods which can be used to make inferences about a population without requiring an
assumption about the distribution of the population are called
a.
continuity-correction methods
b.
non-probabilistic methods
c.
distribution-free methods
d.
non-quantitative methods
47. If the assumption can be made that the populations all have the same shape, the Kruskal-Wallis test
becomes
a.
a test of the skewness of the k populations
b.
a test of the variances of the k populations
c.
a test of the medians of the k populations
d.
a test of the means of the k populations
48. For the Wilcoxon signed-rank test, ties among absolute differences are assigned the
a.
lowest of their ranks
b.
average of their ranks
c.
highest of their ranks
d.
sum of their ranks
49. For the Wilcoxon signed-rank test, differences of 0 are
a.
discarded
b.
assigned a rank of n/2
c.
assigned a rank of n
d.
assigned a rank of n + 1
PROBLEM
1. From the courthouse records, it is found that in 60 divorce cases, the filing for divorce was initiated by
the wife 41 times. Using the sign test, test for a difference in filing between husband and wives. Let
= 0.05
2. Two employers (A and B) ranked five candidates for a new position. Their rankings of the candidates
are shown below.
Rank by A
2
1
3
5
4
Compute the Spearman rank-correlation and test it for significance. Let = 0.05.
3. The following data show the test scores of six individuals on a standardized test before and after
attending a preparation seminar for the test.
Before
108
102
107
97
112
108
Use Wilcoxon Signed-Rank test in order to determine whether or not the seminar has been effective.
Hint: This is a one tailed test. Let = 0.05.
4. Fifteen people were asked to indicate their preference for domestic versus imported cars. The
following data showed their preferences.
Domestic vs. Imported
+
+
−
+
−
−
−
+
+
+
−
+
+
+
−
With = 0.06, test for a significant difference in the preferences for cars. A “+” indicates a preference
for imported cars.
5. The following data show the preference of 20 people for a candidate to a public office. A “+” indicates
a preference for the Democratic candidate, and a “−” indicates a preference for the Republican
candidate.
Republican vs. Democrat
+
−
+
+
+
+
−
+
+
+
+
−
−
+
+
−
−
+
+
+
With = 0.05, test for a significant difference in the preference for the candidates.
6. In a sample of 120 people, 50 indicated that they prefer domestic automobiles, 60 said they prefer
foreign-made cars, and 10 indicated no difference in their preference. At a 0.05 level of significance,
determine if there is evidence of a significant difference in the preferences for the two makes of
automobiles.
7. In a sample of 300 shoppers, 160 indicated they prefer fluoride toothpaste, 120 favored nonfluoride,
and 20 were indifferent. At a 0.05 level of significance, test for a difference in the preference for the
two kinds of toothpaste.
8. Ten administrative assistants were sent to take a typing efficiency course. The following data show the
typing speeds of the administrative assistants before and after the course.
Typing Speed
Before the Course
59
57
60
66
68
59
72
52
58
63
At = 0.05, what can be concluded about the effectiveness of the course?
9. Ten drivers were asked to drive two models of a car. Each car was given one gallon of gasoline. The
distance that each automobile traveled on a gallon of gasoline is shown below.
Distance Traveled (Miles)
Model A
27.7
28.4
28.9
27.9
26.5
29.1
28.9
28.9
28.8
28.0
At = 0.05, what can be concluded about the performance of the two models?
10. The sales records of two branches of a department store over the last 12 months are shown below.
(Sales figures are in thousands of dollars.)
Branch A
257
280
200
250
284
295
297
265
330
350
340
272
Use = 0.05 and test to determine if there is a significant difference in the populations of the sales of
the two branches.
11. Independent random samples of ten day students and ten evening students at a university showed the
following age distributions.
Evening
32
24
23
30
40
41
42
39
45
35
Use = 0.05 and test for any significant differences in the age distribution of the two populations.
12. A PTA group wishes to determine whether a barrage of letters sent to the local station has reduced the
amount of violence broadcast between the hours of 4 and 9 P.M. The results of a survey of viewers are
given here.
Number of Respondents
5
10
6
Carry out a sign test to determine whether or not the letters were effective in reducing the amount of
violence during the 4 to 9 p.m. period. Use a .05 level of significance. Be sure to state the null and
alternative hypotheses.
13. A clothing manufacturer purchased some newly designed sewing machines in the hopes that
production would be increased. The production records of a random sample of workers are shown
below.
Old Machine
28
36
27
25
38
36
40
29
32
28
20
32
32
32
36
Use the Wilcoxon signed-rank test to determine whether the new machines have significantly
increased production. Use the .05 level of significance.
14. The president of a company wants to see if the new anti-smoking campaign is having any influence on
his employees. A sample of 100 employees who smoked prior to the campaign is taken. Thirty-six
employees said they smoked less, 15 employees said they smoked more, and 49 employees said there
was no change.
a.
State the null and alternative hypotheses.
b.
Test the null hypothesis at the 1% level of significance.