7 The Friedman F[sub(r)]-Test for Randomized Block Designs
Tải bản đầy đủ - 0trang
664 ❍
EXAMPLE
CHAPTER 15 NONPARAMETRIC STATISTICS
Test the hypothesis of no association between the populations for Example 15.10.
15.11
Solution The critical value of rs for a one-tailed test with a ϭ .05 and n ϭ 8 is
.643. You may assume that a correlation between the judge’s rank and the teachers’
test scores could not possibly be positive. (A low rank means good teaching and should
be associated with a high test score if the judge and the test measure teaching ability.)
The alternative hypothesis is that the population rank correlation coefficient rs is
less than 0, and you are concerned with a one-tailed statistical test. Thus, a for the
test is the tabulated value for .05, and you can reject the null hypothesis if rs Յ Ϫ.643.
The calculated value of the test statistic, rs ϭ Ϫ.714, is less than the critical value
for a ϭ .05. Hence, the null hypothesis is rejected at the a ϭ .05 level of signiﬁcance. It appears that some agreement does exist between the judge’s rankings and
the test scores. However, it should be noted that this agreement could exist when neither provides an adequate yardstick for measuring teaching ability. For example, the
association could exist if both the judge and those who constructed the teachers’ examination had a completely erroneous, but similar, concept of the characteristics of
good teaching.
What exactly does rs measure? Spearman’s correlation coefficient detects not only
a linear relationship between two variables but also any other monotonic relationship
(either y increases as x increases or y decreases as x increases). For example, if you
calculated rs for the two data sets in Table 15.15, both would produce a value of
rs ϭ 1 because the assigned ranks for x and y in both cases agree for all pairs (x, y).
It is important to remember that a signiﬁcant value of rs indicates a relationship between x and y that is either increasing or decreasing, but is not necessarily linear.
TABLE 15.15
15.8
●
Twin Data Sets With rs ؍1
x
y ϭ x2
1
2
3
4
5
6
1
4
9
16
25
36
y ϭ log10(x)
x
10
100
1000
10,000
100,000
1,000,000
1
2
3
4
5
6
EXERCISES
BASIC TECHNIQUES
15.43 Give the rejection region for a test to detect
positive rank correlation if the number of pairs of
ranks is 16 and you have these a-values:
a. a ϭ .05
b. a ϭ .01
15.44 Give the rejection region for a test to detect
negative rank correlation if the number of pairs of
ranks is 12 and you have these a-values:
a. a ϭ .05
b. a ϭ .01
15.45 Give the rejection region for a test to detect
rank correlation if the number of pairs of ranks is 25
and you have these a-values:
a. a ϭ .05
b. a ϭ .01
15.46 The following paired observations were
obtained on two variables x and y:
x
1.2
.8
2.1
3.5
2.7
1.5
y
1.0
1.3
.1
Ϫ.8
Ϫ.2
.6
15.8 RANK CORRELATION COEFFICIENT
a. Calculate Spearman’s rank correlation coefficient
rs.
b. Do the data present sufficient evidence to indicate a
correlation between x and y? Test using a ϭ .05.
15.47 Rating Political Candidates A
political scientist wished to examine the relationship between the voter image of a conservative
political candidate and the distance (in miles) between
the residences of the voter and the candidate. Each of
12 voters rated the candidate on a scale of 1 to 20.
Voter
Rating
Distance
1
2
3
4
5
6
7
8
9
10
11
12
12
7
5
19
17
12
9
18
3
8
15
4
75
165
300
15
180
240
120
60
230
200
130
130
a. Calculate Spearman’s rank correlation coefficient
rs.
b. Do these data provide sufficient evidence to indicate a negative correlation between rating and
distance?
15.48 Competitive Running Is the number
of years of competitive running experience
related to a runner’s distance running performance?
The data on nine runners, obtained from the study by
Scott Powers and colleagues, are shown in the table:5
EX1548
Runner
Years of Competitive
Running
10-Kilometer
Finish Time (min)
1
2
3
4
5
6
7
8
9
9
13
5
7
12
6
4
5
3
33.15
33.33
33.50
33.55
33.73
33.86
33.90
34.15
34.90
a. Calculate the rank correlation coefficient between
years of competitive running x and a runner’s ﬁnish
time y in the 10-kilometer race.
b. Do the data provide sufficient evidence to indicate a
rank correlation between y and x? Test using a ϭ .05.
665
15.49 Tennis Racquets The data shown in
the accompanying table give measures of
bending stiffness and twisting stiffness as determined
by engineering tests on 12 tennis racquets.
EX1549
Racquet
Bending
Stiffness
Twisting
Stiffness
1
2
3
4
5
6
7
8
9
10
11
12
419
407
363
360
257
622
424
359
346
556
474
441
227
231
200
211
182
304
384
194
158
225
305
235
APPLICATIONS
EX1547
❍
a. Calculate the rank correlation coefficient rs between
bending stiffness and twisting stiffness.
b. If a racquet has bending stiffness, is it also likely to
have twisting stiffness? Use the rank correlation
coefficient to determine whether there is a signiﬁcant positive relationship between bending stiffness
and twisting stiffness. Use a ϭ .05.
15.50 Student Ratings A school principal sus-
pected that a teacher’s attitude toward a ﬁrst-grader
depended on his original judgment of the child’s ability. The principal also suspected that much of that
judgment was based on the ﬁrst-grader’s IQ score,
which was usually known to the teacher. After three
weeks of teaching, a teacher was asked to rank the
nine children in his class from 1 (highest) to 9 (lowest)
as to his opinion of their ability. Calculate rs for these
teacher–IQ ranks:
Teacher
1
2
3
4
5
6
7
8
9
IQ
3
1
2
4
5
7
9
6
8
15.51 Student Ratings, continued Refer to Exercise 15.50. Do the data provide sufficient evidence to
indicate a positive correlation between the teacher’s
ranks and the ranks of the IQs? Use a ϭ .05.
15.52 Art Critics Two art critics each ranked
10 paintings by contemporary (but anonymous)
artists in accordance with their appeal to the respective
critics. The ratings are shown in the table. Do the critics
seem to agree on their ratings of contemporary art? That
is, do the data provide sufficient evidence to indicate a
positive correlation between critics A and B? Test by
using a value of a near .05.
EX1552
666
❍
CHAPTER 15 NONPARAMETRIC STATISTICS
Painting
Critic A
Critic B
1
2
3
4
5
6
7
8
9
10
6
4
9
1
2
7
3
8
5
10
5
6
10
2
3
8
1
7
4
9
Calculate rs. Do the data provide sufficient evidence to
indicate an association between the grader’s ratings
and the moisture contents of the leaves?
15.54 Social Skills Training A social skills
training program was implemented with seven
mildly challenged students in a study to determine
whether the program caused improvements in pre/post
measures and behavior ratings. For one such test, the
pre- and posttest scores for the seven students are
given in the table:
EX1554
15.53 Rating Tobacco Leaves An experi-
ment was conducted to study the relationship
between the ratings of a tobacco leaf grader and the
moisture content of the tobacco leaves. Twelve leaves
were rated by the grader on a scale of 1 to 10, and corresponding readings of moisture content were made.
Student
Pretest
Posttest
Earl
Ned
Jasper
Charlie
Tom
Susie
Lori
101
89
112
105
90
91
89
113
89
121
99
104
94
99
EX1553
Leaf
Grader’s Rating
Moisture Content
1
2
3
4
5
6
7
8
9
10
11
12
9
6
7
7
5
8
2
6
1
10
9
3
.22
.16
.17
.14
.12
.19
.10
.12
.05
.20
.16
.09
15.9
a. Use a nonparametric test to determine whether there
is a signiﬁcant positive relationship between the
pre- and posttest scores.
b. Do these results agree with the results of the parametric test in Exercise 12.51?
SUMMARY
The nonparametric tests presented in this chapter are only a few of the many nonparametric tests available to experimenters. The tests presented here are those for
which tables of critical values are readily available.
Nonparametric statistical methods are especially useful when the observations can
be rank ordered but cannot be located exactly on a measurement scale. Also, nonparametric methods are the only methods that can be used when the sampling designs
have been correctly adhered to, but the data are not or cannot be assumed to follow
the prescribed one or more distributional assumptions.
We have presented a wide array of nonparametric techniques that can be used when
either the data are not normally distributed or the other required assumptions are not
met. One-sample procedures are available in the literature; however, we have concentrated on analyzing two or more samples that have been properly selected using
random and independent sampling as required by the design involved. The nonparametric analogues of the parametric procedures presented in Chapters 10–14 are
straightforward and fairly simple to implement:
•
•
The Wilcoxon rank sum test is the nonparametric analogue of the two-sample
t-test.
The sign test and the Wilcoxon signed-rank tests are the nonparametric
analogues of the paired-sample t-test.
CHAPTER REVIEW
•
•
•
❍
667
The Kruskal–Wallis H-test is the rank equivalent of the one-way analysis of
variance F-test.
The Friedman Fr-test is the rank equivalent of the randomized block design
two-way analysis of variance F-test.
Spearman’s rank correlation rs is the rank equivalent of Pearson’s correlation
coefficient.
These and many more nonparametric procedures are available as alternatives to
the parametric tests presented earlier. It is important to keep in mind that when the
assumptions required of the sampled populations are relaxed, our ability to detect
signiﬁcant differences in one or more population characteristics is decreased.
CHAPTER REVIEW
Key Concepts and Formulas
I.
Nonparametric Methods
1. These methods can be used when the data cannot be measured on a quantitative scale, or
when
2. The numerical scale of measurement is arbitrarily set by the researcher, or when
3. The parametric assumptions such as normality
or constant variance are seriously violated.
II. Wilcoxon Rank Sum Test: Independent Random Samples
1. Jointly rank the two samples. Designate the
smaller sample as sample 1. Then
III. Sign Test for a Paired Experiment
1. Find x, the number of times that observation A
exceeds observation B for a given pair.
2. To test for a difference in two populations, test
H0 : p ϭ .5 versus a one- or two-tailed alternative.
3. Use Table 1 of Appendix I to calculate the
p-value for the test.
4. When the sample sizes are large, use the normal approximation:
x Ϫ .5n
z ϭ ᎏᎏ
.5͙nෆ
T1 ϭ Rank sum of sample 1
T *1 ϭ n1(n1 ϩ n2 ϩ 1) Ϫ T1
2. Use T1 to test for population 1 to the left of
population 2. Use T *1 to test for population 1 to
the right of population 2. Use the smaller of T1
and T *1 to test for a difference in the locations
of the two populations.
3. Table 7 of Appendix I has critical values for
the rejection of H0.
4. When the sample sizes are large, use the normal approximation:
n1(n1 ϩ n2 ϩ 1)
mT ϭ ᎏ
ᎏ
2
n1n2(n1 ϩ n2 ϩ 1)
s 2T ϭ ᎏ
ᎏ
12
T Ϫ mT
z ϭ ᎏᎏ
sT
IV. Wilcoxon Signed-Rank Test: Paired
Experiment
1. Calculate the differences in the paired observations. Rank the absolute values of the differences. Calculate the rank sums T ϩ and T Ϫ for
the positive and negative differences, respectively. The test statistic T is the smaller of the
two rank sums.
2. Table 8 in Appendix I has critical values for
the rejection of H0 for both one- and two-tailed
tests.
3. When the sample sizes are large, use the normal approximation:
T ϩ Ϫ [n(n ϩ 1)/4]
z ϭ ᎏᎏᎏ
ෆ
[n(n ϩෆϩ
1)(2n ෆ
1)]/24
͙ෆ
668 ❍
CHAPTER 15 NONPARAMETRIC STATISTICS
V. Kruskal–Wallis H-Test: Completely
Randomized Design
1. Jointly rank the n observations in the k samples. Calculate the rank sums, Ti ϭ rank sum of
sample i, and the test statistic
12
T2
H ϭ ᎏᎏ S ᎏᎏi Ϫ 3(n ϩ 1)
n(n ϩ 1)
ni
2. If the null hypothesis of equality of distributions is false, H will be unusually large, resulting in a one-tailed test.
3. For sample sizes of ﬁve or greater, the rejection region for H is based on the chi-square
distribution with (k Ϫ 1) degrees of freedom.
VI. The Friedman Fr-Test: Randomized
Block Design
1. Rank the responses within each block from 1 to
k. Calculate the rank sums, T1, T2, . . . , Tk, and
the test statistic
12
Fr ϭ ᎏᎏ S T 2i Ϫ 3b(k ϩ 1)
bk(k ϩ 1)
3. For block sizes of ﬁve or greater, the rejection region for Fr is based on the chi-square
distribution with (k Ϫ 1) degrees of freedom.
VII. Spearman’s Rank Correlation
Coefficient
1. Rank the responses for the two variables from
smallest to largest.
2. Calculate the correlation coefficient for the
ranked observations:
Sxy
rs ϭ ᎏᎏ or
ෆ
͙S
xxSyy
6 S d 2i
rs ϭ 1 Ϫ ᎏ
ᎏ
n(n2 Ϫ 1)
if there are no ties
3. Table 9 in Appendix I gives critical values for
rank correlations signiﬁcantly different
from 0.
4. The rank correlation coefficient detects not
only signiﬁcant linear correlation but also any
other monotonic relationship between the two
variables.
2. If the null hypothesis of equality of treatment
distributions is false, Fr will be unusually large,
resulting in a one-tailed test.
Nonparametric Procedures
Many nonparametric procedures are available in the MINITAB package, including most
of the tests discussed in this chapter. The Dialog boxes are all familiar to you by now,
and we will discuss the tests in the order presented in the chapter.
To implement the Wilcoxon rank sum test for two independent random samples,
enter the two sets of sample data into two columns (say, C1 and C2) of the MINITAB
worksheet. The Dialog box in Figure 15.13 is generated using Stat Ǟ Nonparametrics Ǟ Mann-Whitney. Select C1 and C2 for the First and Second Samples, and
indicate the appropriate conﬁdence coefficient (for a conﬁdence interval) and alternative hypothesis. Clicking OK will generate the output in Figure 15.1.
The sign test and the Wilcoxon signed-rank test for paired samples are performed
in exactly the same way, with a change only in the last command of the sequence.
Even the Dialog boxes are identical! Enter the data into two columns of the MINITAB
worksheet (we used the cake mix data in Section 15.5). Before you can implement
either test, you must generate a column of differences using Calc Ǟ Calculator, as
shown in Figure 15.14. Use Stat Ǟ Nonparametrics Ǟ 1-Sample Sign or Stat Ǟ
Nonparametrics Ǟ 1-Sample Wilcoxon to generate the appropriate Dialog box
MY MINITAB
FIGU R E 1 5 . 1 3
●
FIGU R E 1 5 . 1 4
●
❍
669
shown in Figure 15.15. Remember that the median is the value of a variable such that
50% of the values are smaller and 50% are larger. Hence, if the two population distributions are the same, the median of the differences will be 0. This is equivalent to
the null hypothesis
H0 : P(positive difference) ϭ P(negative difference) ϭ .5
670 ❍
CHAPTER 15 NONPARAMETRIC STATISTICS
FI GU R E 1 5 . 1 5
●
used for the sign test. Select the column of differences for the Variables box, and select the test of the median equals 0 with the appropriate alternative. Click OK to obtain the printout for either of the two tests. The Session window printout for the sign
test, shown in Figure 15.16, indicates a nonsigniﬁcant difference in the distributions
of densities for the two cake mixes. Notice that the p-value (.2188) is not the same as
the p-value for the Wilcoxon signed-rank test (.093 from Figure 15.4). However, if
you are testing at the 5% level, both tests produce nonsigniﬁcant differences.
FI GU R E 1 5 . 1 6
●
The procedures for implementing the Kruskal–Wallis H-test for k independent samples and Friedman’s Fr-test for a randomized block design are identical to the procedures used for their parametric equivalents. Review the methods described in the
section “MyMINITAB ” in Chapter 11. Once you have entered the data as explained in
that section, the commands Stat Ǟ Nonparametrics Ǟ Kruskal–Wallis or Stat Ǟ
Nonparametrics Ǟ Friedman will generate a Dialog box in which you specify the
Response column and the Factor column, or the response column, the treatment column and the block column, respectively. Click OK to obtain the outputs for these
nonparametric tests.
SUPPLEMENTARY EXERCISES
❍
671
Finally, you can generate the nonparametric rank correlation coefficient rs if you
enter the data into two columns and rank the data using Data Ǟ Rank. For example,
the data on judge’s rank and test scores were entered into columns C6 and C7 of our
MINITAB worksheet. Since the judge’s ranks are already in rank order, we need only
to rank C7 by selecting “Exam Score” and storing the ranks in C8 [named “Rank ( y)”
in Figure 15.17]. The commands Stat Ǟ Basic Statistics Ǟ Correlation will now
produce the rank correlation coefficient when C6 and C8 are selected. However, the
p-value that you see in the output does not produce exactly the same test as the critical values in Table 15.14. You should compare your value of rs with the tabled value
to check for a signiﬁcant association between the two variables.
FIGU R E 1 5 . 1 7
●
Supplementary Exercises
15.55 Response Times An experiment was
conducted to compare the response times for
two different stimuli. To remove natural person-toperson variability in the responses, both stimuli were
presented to each of nine subjects, thus permitting an
analysis of the differences between stimuli within each
person. The table lists the response times (in seconds).
EX1555
Subject
Stimulus 1
Stimulus 2
1
2
3
4
5
6
7
8
9
9.4
7.8
5.6
12.1
6.9
4.2
8.8
7.7
6.4
10.3
8.9
4.1
14.7
8.7
7.1
11.3
5.2
7.8
a. Use the sign test to determine whether sufficient
evidence exists to indicate a difference in the mean
response times for the two stimuli. Use a rejection
region for which a Յ .05.
b. Test the hypothesis of no difference in mean
response times using Student’s t-test.
15.56 Response Times, continued Refer to Exercise 15.55. Test the hypothesis that no difference
exists in the distributions of response times for the two
stimuli, using the Wilcoxon signed-rank test. Use a
rejection region for which a is as near as possible to
the a achieved in Exercise 15.55, part a.
15.57 Identical Twins To compare two
junior high schools, A and B, in academic
effectiveness, an experiment was designed requiring
EX1557
672
❍
CHAPTER 15 NONPARAMETRIC STATISTICS
the use of 10 sets of identical twins, each twin having
just completed the sixth grade. In each case, the twins
in the same set had obtained their schooling in the
same classrooms at each grade level. One child was
selected at random from each pair of twins and
assigned to school A. The remaining children were
sent to school B. Near the end of the ninth grade, a
certain achievement test was given to each child in the
experiment. The test scores are shown in the table.
Twin Pair
School A
School B
1
2
3
4
5
6
7
8
9
10
67
80
65
70
86
50
63
81
86
60
39
75
69
55
74
52
56
72
89
47
15.58 Identical Twins II Refer to Exercise 15.57.
What answers are obtained if Wilcoxon’s signed-rank
test is used in analyzing the data? Compare with your
earlier answers.
15.59 Paper Brightness The coded values
for a measure of brightness in paper (light
EX1559
reﬂectivity), prepared by two different processes, are
given in the table for samples of nine observations
drawn randomly from each of the two processes. Do
the data present sufficient evidence to indicate a difference in the brightness measurements for the two processes? Use both a parametric and a nonparametric test
and compare your results.
A
B
Brightness
6.1
9.1
9.2
8.2
8.7
8.6
8.9
6.9
7.6
7.5
b. Number the ranked observations “from the outside
in”; that is, number the smallest observation 1, the
largest 2, the next-to-smallest 3, the next-to-largest
4, and so on. This sequence of numbers induces an
ordering on the symbols A (population A items)
and B (population B items). If s 2A Ͼ s 2B, one
would expect to ﬁnd a preponderance of A’s near
the ﬁrst of the sequences, and thus a relatively
small “sum of ranks” for the A observations.
c. Given the measurements in the table produced by
well-calibrated precision instruments A and B, test
at near the a ϭ .05 level to determine whether the
more expensive instrument B is more precise than
A. (Note that this implies a one-tailed test.) Use
the Wilcoxon rank sum test statistic.
a. Test (using the sign test) the hypothesis that the
two schools are the same in academic effectiveness, as measured by scores on the achievement
test, versus the alternative that the schools are not
equally effective.
b. Suppose it was known that junior high school A
had a superior faculty and better learning facilities.
Test the hypothesis of equal academic effectiveness
versus the alternative that school A is superior.
Process
a. Rank the combined sample.
7.1
7.9
9.5
8.3
8.3
7.8
9.0
8.9
15.60 Precision Instruments Assume (as in the
case of measurements produced by two well-calibrated
measuring instruments) the means of two populations
are equal. Use the Wilcoxon rank sum statistic for testing hypotheses concerning the population variances as
follows:
Instrument A
Instrument B
1060.21
1060.34
1060.27
1060.36
1060.40
1060.24
1060.28
1060.32
1060.30
d. Test using the equality of variance F-test.
15.61 Meat Tenderizers An experiment
was conducted to compare the tenderness of
meat cuts treated with two different meat tenderizers, A
and B. To reduce the effect of extraneous variables, the
data were paired by the speciﬁc meat cut, by applying
the tenderizers to two cuts taken from the same steer,
by cooking paired cuts together, and by using a single
judge for each pair. After cooking, each cut was rated
by a judge on a scale of 1 to 10, with 10 corresponding
to the most tender meat. The data are shown for a single judge. Do the data provide sufficient evidence to
indicate that one of the two tenderizers tends to receive
higher ratings than the other? Would a Student’s t-test
be appropriate for analyzing these data? Explain.
EX1561
Tenderizer
Cut
A
B
Shoulder roast
Chuck roast
Rib steak
Brisket
Club steak
Round steak
Rump roast
Sirloin steak
Sirloin tip steak
T-bone steak
5
6
8
4
9
3
7
8
8
9
7
5
9
5
9
5
6
8
9
10
15.62 Interviewing Job Prospects A large
EX1562
corporation selects college graduates for
SUPPLEMENTARY EXERCISES
employment using both interviews and a psychological
achievement test. Interviews conducted at the home
office of the company are far more expensive than the
tests that can be conducted on campus. Consequently,
the personnel office was interested in determining
whether the test scores were correlated with interview
ratings and whether tests could be substituted for interviews. The idea was not to eliminate interviews but to
reduce their number. To determine whether the measures were correlated, 10 prospects were ranked
during interviews and tested. The paired scores are as
listed here:
Subject
Interview Rank
Test Score
1
2
3
4
5
6
7
8
9
10
8
5
10
3
6
1
4
7
9
2
74
81
66
83
66
94
96
70
61
86
15.63 Interviews, continued Refer to Exercise
15.62. Do the data present sufficient evidence to indicate that the correlation between interview rankings
and test scores is less than zero? If this evidence does
exist, can you say that tests can be used to reduce the
number of interviews?
15.64 Word Association Experiments A compari-
son of reaction times for two different stimuli in a psychological word-association experiment produced the
accompanying results when applied to a random sample of 16 people:
1
2
Reaction Time (sec)
1
4
3
2
2
3
1
3
2
1
673
Student
Math
Art
Student
Math
Art
1
2
3
4
5
6
7
8
22
37
36
38
42
58
58
60
53
68
42
49
51
65
51
71
9
10
11
12
13
14
15
62
65
66
56
66
67
62
55
74
68
64
67
73
65
15.66 Math and Art, continued Refer to Exercise 15.65. Compute Spearman’s rank correlation
coefficient for these data and test H0 : no association
between the rank pairs at the 10% level of signiﬁcance.
15.67 Yield of Wheat Exercise 11.68 presented an
analysis of variance of the yields of ﬁve different varieties of wheat, observed on one plot each at each of
six different locations (see data set EX1168). The data
from this randomized block design are listed here:
Location
Calculate the Spearman rank correlation coefficient rs.
Rank 1 is assigned to the candidate judged to be the
best.
Stimulus
❍
1
2
3
3
2
3
Varieties
1
2
3
4
5
6
A
B
C
D
E
35.3
30.7
38.2
34.9
32.4
31.0
32.2
33.4
36.1
28.9
32.7
31.4
33.6
35.2
29.2
36.8
31.7
37.1
38.3
30.7
37.2
35.0
37.3
40.2
33.9
33.1
32.7
38.2
36.0
32.1
a. Use the appropriate nonparametric test to determine whether the data provide sufficient evidence
to indicate a difference in the yields for the ﬁve
different varieties of wheat. Test using a ϭ .05.
b. Exercise 11.68 presented a computer printout of
the analysis of variance for comparing the mean
yields for the ﬁve varieties of wheat. How do the
results of the analysis of variance F test compare
with the test in part a? Explain.
15.68 Learning to Sell In Exercise 11.61, you com-
pared the numbers of sales per trainee after completion
of one of four different sales training programs (see
data set EX1161). Six trainees completed training program 1, eight completed program 2, and so on. The
numbers of sales per trainee are shown in the table.
Training Program
Do the data present sufficient evidence to indicate a
difference in mean reaction times for the two stimuli?
Use an appropriate nonparametric test and explain
your conclusions.
1
15.65 Math and Art The table gives the
scores of a group of 15 students in mathematics
and art. Use Wilcoxon’s signed-rank test to determine
whether the median scores for these students differ
signiﬁcantly for the two subjects.
EX1565
Total
2
3
4
78
84
86
92
69
73
99
86
90
93
94
85
97
91
74
87
80
83
78
81
63
71
65
86
79
73
70
482
735
402
588
CHAPTER 15 NONPARAMETRIC STATISTICS
b. How do the test results in part a compare with the
results of the analysis of variance F-test in Exercise 11.61?
15.69 Pollution from Chemical Plants In Exercise 11.66, you performed an analysis of variance to
compare the mean levels of effluents in water at four
different industrial plants (see data set EX1166). Five
samples of liquid waste were taken at the output of
each of four industrial plants. The data are shown in
the table.
Plant
A
B
C
D
Polluting Effluents (lb/gal of waste)
1.65
1.70
1.40
2.10
1.72
1.85
1.75
1.95
1.50
1.46
1.38
1.65
1.37
2.05
1.65
1.88
1.60
1.80
1.55
2.00
a. Do the data present sufficient evidence to indicate
a difference in the levels of pollutants for the four
different industrial plants? Test using the appropriate nonparametric test.
b. Find the approximate p-value for the test and interpret its value.
c. Compare the test results in part a with the analysis
of variance test in Exercise 11.66. Do the results
agree? Explain.
15.70 AIDS Research Scientists have shown that a
newly developed vaccine can shield rhesus monkeys
from infection by a virus closely related to the AIDScausing human immunodeﬁciency virus (HIV). In their
work, Ronald C. Resrosiers and his colleagues at the
New England Regional Primate Research Center gave
each of n ϭ 6 rhesus monkeys ﬁve inoculations with
the simian immunodeﬁciency virus (SIV) vaccine. One
week after the last vaccination, each monkey received
an injection of live SIV. Two of the six vaccinated
monkeys showed no evidence of SIV infection for as
long as a year and a half after the SIV injection.6 Scientists were able to isolate the SIV virus from the
other four vaccinated monkeys, although these animals
showed no sign of the disease. Does this information
contain sufficient evidence to indicate that the vaccine
is effective in protecting monkeys from SIV? Use
a ϭ .10.
15.71 Heavy Metal An experiment was
EX1571
performed to determine whether there is an
accumulation of heavy metals in plants that were
grown in soils amended with sludge and whether there
is an accumulation of heavy metals in insects feeding
on those plants.7 The data in the table are cadmium
concentrations (in mg/kg) in plants grown under six
different rates of application of sludge for three different harvests. The rates of application are the treatments. The three harvests represent time blocks in the
two-way design.
Harvest
Rate
1
2
3
Control
1
2
3
4
5
162.1
199.8
220.0
194.4
204.3
218.9
153.7
199.6
210.7
179.0
203.7
236.1
200.4
278.2
294.8
341.1
330.2
344.2
a. Based on the MINITAB normal probability plot and
the plot of residuals versus rates, are you willing to
assume that the normality and constant variance assumptions are satisﬁed?
MINITAB residual plots for Exercise 15.71
Residuals versus Rate
(response is Cadmium)
40
30
20
Residual
a. Do the data present sufficient evidence to indicate
that the distribution of number of sales per trainee
differs from one training program to another? Test
using the appropriate nonparametric test.
10
0
Ϫ10
Ϫ20
Ϫ30
Ϫ40
0
1
2
3
4
5
Rate
Normal Probability Plot of the Residuals
(response is Cadmium)
99
95
90
Percent
674 ❍
80
70
60
50
40
30
20
10
5
1
Ϫ40
Ϫ30
Ϫ20
Ϫ10
0
10
Residual
20
30
40
50