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7 The Friedman F[sub(r)]-Test for Randomized Block Designs

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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

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

2

n1n2(n1 ϩ n2 ϩ 1)

s 2T ϭ 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

[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

͙S

xxSyy

6 S d 2i

rs ϭ 1 Ϫ 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

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

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

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

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

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

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

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