16 - 1 copyright © 2004 by the mcgraw-hill companies, inc. all rights reserved

16 - 1

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

16 - 2


1.1. Conduct the sign test for single and dependent samples using the binomial and standard normal distributions

as the test statistics

2.2. Conduct a test of hypothesis for dependent samples using the Wilcoxon signed-rank test

When you have completed this chapter, you will be able to:

3.3. Conduct and interpret the Wilcoxon rank-sum test for independent samples

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4.4.

5.5.

6.6.

Conduct and interpret the Kruskal-Wallis test

for several independent samples

Compute and interpret Spearman’s coefficient of rank correlation

Conduct a test of hypothesis to determine whether the correlation among the ranks in the population is

different from zero

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TerminologyRange

…is the difference between the largest and the smallest value

…is the difference between the largest and the smallest value

Only two values are used in its calculation It is influenced by an extreme value It is easy to compute and understand

Only two values are used in its calculation It is influenced by an extreme value It is easy to compute and understand

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The Sign TestThe Sign Test

The Sign Test is based on the sign of a difference between two related observations

... no assumption is necessary regarding the shape of the population of differences

… the binomial distribution is the test statistic for small samples and the standard normal (z) for

large samples

… the test requires dependent (related) samples

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Determine the sign of the difference

between related pairs

Determine the sign of the difference

between related pairs1. Determine the number of usable pairs

2. Compare the number of positive (or negative) differences to the critical value

3. n is the number of usable pairs (without ties), x is the number of pluses or minuses,

and the binomial probability p=.5

The Sign TestThe Sign Test …continued

Procedure to conduct the test:

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…if the number of pluses or minuses is more than n/2, then

…if the number of pluses or minuses is less than n/2, then

Normal ApproximationNormal Approximation

z x n

n

( . ) .

.

5 5

5

z x n

n

( . ) .

.

5 5

5

…if both and are greater than 5,

the z distribution is appropriate

np n( )1 p

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The Gagliano Research Institute for Business Studies is comparing the Research and Development expense (R&D) as a percent of income for a sample of glass

manufacturing firms for 2000 and 2001

At the .05 significance level has the R&D expense declined?

Use the sign test


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Company 2000 2001 Difference Sign

Savoth Glass 20 16 4 +

Ruisi Glass 14 13 1 +

Rubin Inc.. 23 20 3 +

Vaught 24 17 7 +

Lambert Glass 31 22 9 +

Pimental 22 20 2 +

Olson Glass 14 20 - 6 -

Flynn Glass 18 11 7 +

Company 2000 2001 Difference Sign

Savoth Glass 20 16 4 +

Ruisi Glass 14 13 1 +

Rubin Inc.. 23 20 3 +

Vaught 24 17 7 +

Lambert Glass 31 22 9 +

Pimental 22 20 2 +

Olson Glass 14 20 - 6 -

Flynn Glass 18 11 7 +


Continued…

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Step 1: H0: p=.5 H1: p<.5

Step 2: H0: is rejected if the number of negative signs is 0 or 1

Step 3: There is one negative difference, i.e. there was an increase in the percent for one company

Step 4: H0 is rejected

R&D expense as a percent of income declined from 2000 to 2001


Continued…

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Testing a Hypothesis About a Median

Testing a Hypothesis About a Median

When testing the value of the median, we use the

normal approximation to the binomial

distribution

When testing the value of the median, we use the

normal approximation to the binomial

distribution The z distribution is used as the test statistic

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The Gordon Travel Agency claims that their median airfare for all their clients to all

destinations is $450.

This claim is being challenged by a competing agency, who believe the

median is different from $450.

A random sample of 300 tickets revealed 170 tickets were below $450.

Use the 0.05 level of significance.

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Continued…

H0 is rejected if z is > than –1.96 or < than 1.96

… the value of z is 2.252

H0 is rejected. We conclude that the median is not

$450

450$ median :H $450 =median : 10 H

252.23005.

)300(50.)5.170(

5.50.)5.(

nnxz

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Wilcoxon Signed-Rank Test

… the test requires the ordinal scale of measurement

… the observations must be related or dependent

Use the Wilcoxon Signed-Rank if the assumption of normality is violated for the paired-t test

Note

ontinued…

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Steps 1. Compute the differences between related observations

2. Rank the absolute differences from low to high

3. Return the signs to the ranks and sum positive and negative ranks

4. Compare the smaller of the two rank sums

with the T value, obtained from the Appendix

of Wilcoxon T values.

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Use the Wilcoxon matched-pair signed-rank test to determine if the R&D expenses as a percent of

income (EXAMPLE 1) have declined. Use the .05 significance level

Step 1: H0: The percent stayed the same H1: The percent declined

Step 2: H0 is rejected if the smaller of the rank

sums is less than or equal to 5 See the Appendix of Wilcoxon T Values

ontinued…

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Company 2000 2001 Difference ABS-Diff Rank R+ R-

Savoth Glass 20 16 4 4 4 4 *

Ruisi Glass 14 13 1 1 1 1 *

Rubin Inc.. 23 20 3 3 3 3 *

Vaught 24 17 7 7 7 7 *

Lambert Glass 31 22 9 8 8 8 *

Pimental 22 20 2 2 2 2 *

Olson Glass 14 20 - 6 6 5 * 5

Flynn Glass 18 11 7 7 6 6 *ontinued…


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The smaller rank sum is 5, which is equal to the critical value of T.

H0 is rejectedThe percent has declined from one year to the next.

Company 2000 2001 Difference ABS-Diff Rank R+ R-Savoth Glass 20 16 4 4 4 4 *

Ruisi Glass 14 13 1 1 1 1 *Rubin Inc.. 23 20 3 3 3 3 *

Vaught 24 17 7 7 7 7 *Lambert Glass 31 22 9 8 8 8 *Pimental 22 20 2 2 2 2 *Olson Glass 14 20 - 6 6 5 * 5

Flynn Glass 18 11 7 7 6 6 *


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...used to determine if two independent samples came from the same or equal populations

…no assumption about the shape of the population is required

…the data must be at least ordinal scale

…each sample must contain at least eight observations

Wilcoxon Rank-Sum Test

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…to determine the value of the test statistic W, all data values are ranked from low to high

as if they were from a single population

…the sum of ranks for each of the two samples is determined


The smaller of the two sums W is used to compute the test statistic from:

12

)1(2

)1(

2121

211

nnnn

nnnW

z

16 - 21


Your school purchased two vehicles, a Ford and a Chevrolet, for the

administration’s use when travelling.

The repair costs for the two cars over the last three years is shown

on the next slide.

At the .05 significance level is there a difference in the two distributions?


ontinued…

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Ford ($) Rank Chevrolet($) Rank

25.31 3.0 14.89 1.0

33.68 5.5 20.31 2.0

46.89 7.0 25.97 4.0

51.83 8.0 33.68 5.5

87.65 13.0 68.98 9.0

87.90 14.0 78.23 10.0

90.89 15.0 80.31 11.0

120.67 16.0 81.75 12.0

157.90 17.0

81.5 71.5


ontinued…

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Step 1: H0: The percent stayed the same H1: The percent declined

Step 2: H0 is rejected if z >1.96 or z is less than –1.96


Step 3: The value of the test statistic is 0.914.

12

)198)(9(82

)198(85.81

12

)1(2

)1(

2121

211

nnnn

nnnW

z= 0.914= 0.914

ontinued…

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Step 4: The value of the test statistic is 0.914.

We do not reject the null hypothesis!

We cannot conclude that there is a

difference in the distributions of the repair costs

of the two vehicles!

We cannot conclude that there is a

difference in the distributions of the repair costs

of the two vehicles!

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Kruskal-Wallis Test: Analysis of Variance by

Ranks


Ranks…used to compare three or more samples

to determine if they came from equal populations

…the ordinal scale of measurement is required

…it is an alternative to the one-way ANOVA

…the chi-square distribution is the test statistic

…each sample should have at least five observations

…the sample data is ranked from low to high as if it were a single group

Note

ontinued…

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The test statistic is given by:


Ranks


Rankscontinued…

)HN N

R

n

R

n

R

nN

k

k

12

13 1

12

1

22

2

2

( )

( ) ( )...

( )(

16 - 27


Keely Ambrose, director of Human Resources for Miller Industries, wishes to study the

percent increase in salary for middle managers at the four manufacturing plants. She gathers a sample of managers and determines the percent increase in salary

from last year to this year. At the 5% significance level,

can Keely conclude that there is a difference in the percent increases for the

various plants?


Ranks


Ranks

ontinued…

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Ranks


Ranks

Millville Rank Camden Rank Eaton Rank Cliff Rank

2.2 2.0 1.9 1 3.7 6.0 5.7 9.0

3.6 5.0 2.7 3 4.5 7.0 6.8 10.5

4.9 8.0 3.1 4 7.1 13.5 8.9 16.0

6.8 10.5 6.9 12 9.3 17.0 11.6 18.5

7.1 13.5 8.3 15 11.6 18.5 13.9 20.0

39.0 35 62.0 74.0

Millville Rank Camden Rank Eaton Rank Cliff Rank

2.2 2.0 1.9 1 3.7 6.0 5.7 9.0

3.6 5.0 2.7 3 4.5 7.0 6.8 10.5

4.9 8.0 3.1 4 7.1 13.5 8.9 16.0

6.8 10.5 6.9 12 9.3 17.0 11.6 18.5

7.1 13.5 8.3 15 11.6 18.5 13.9 20.0

39.0 35 62.0 74.0

PlantsPlants

ontinued…

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Step 1: H0: The populations are the same H1: The populations are not the same

Step 2: H0 is rejected if x2 is greater than 7.185.

There are 3 degrees of freedom at the .05 significance level.


Ranks


Ranks

ontinued…

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)120(35

74

5

62

5

35

5

39

)120(20

12 2222

)1(3)()()()(

)1(12 2

4

2

23

2

22

1

21

N

n

R

n

R

n

R

n

R

NNH

k

= 5.949= 5.949


Ranks


RanksMillville Rank Camden Rank Eaton Rank Cliff Rank 2.2 2.0 1.9 1 3.7 6.0 5.7 9.0 3.6 5.0 2.7 3 4.5 7.0 6.8 10.5 4.9 8.0 3.1 4 7.1 13.5 8.9 16.0 6.8 10.5 6.9 12 9.3 17.0 11.6 18.5 7.1 13.5 8.3 15 11.6 18.5 13.9 20.0

39.0 35 62.0 74.0

ontinued…

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The null hypothesis is not rejected


Ranks


Ranks

There is no difference in the percent increases in the four plants

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Rank-Order CorrelationRank-Order Correlation

…Spearman’s coefficient of rank correlation reports the association between two sets of

ranked observations

Features

…it can range from –1.00 up to 1.00

…it is similar to Pearson’s coefficient of correlation, but is based on ranked data

ontinued…

16 - 33


Spearman’s Rank-Order Correlation


Formula (to find the coefficient of rank correlation)

d is the difference in the ranks and n is the number of observations.

rd

n ns 1

6

1

2

2

( )

ontinued…

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Testing the significance of rs

State the null hypothesis: Rank correlation in population is 0

State the alternate hypothesis:

Rank correlation in population is not 0.

The value of the test statistic is computed from …

t rs

nrs

s

21

16 - 35




The preseason football rankings for the Atlantic Coast Conference by the coaches and sports writers

are shown below. What is the coefficient of rank correlation between the two groups?

School Coaches WritersMaryland 2 3NC State 3 4NC 6 6Virginia 5 5Clemson 4 2Wake Forest 7 8Duke 8 7Florida State 1 1

School Coaches WritersMaryland 2 3NC State 3 4NC 6 6Virginia 5 5Clemson 4 2Wake Forest 7 8Duke 8 7Florida State 1 1 ontinued…

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School Coaches Writers d d2

Maryland 2 3 -1 1

NC State 3 4 -1 1NC 6 6 0 0Virginia 5 5 0 0Clemson 4 2 2 4Wake Forest 7 8 -1 1Duke 8 7 1 1Florida State 1 1 0 0



Total 8Total 8

ontinued…

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School Coaches Writers d d2

Maryland 2 3 -1 1

NC State 3 4 -1 1NC 6 6 0 0Virginia 5 5 0 0Clemson 4 2 2 4Wake Forest 7 8 -1 1Duke 8 7 1 1Florida State 1 1 0 0Total 8Total 8

= 0.905= 0.905)18(8

)8(61 2

)1(

61 2

2

nn

drs

There is a strong correlation between the ranks of the coaches and the sports writers!

There is a strong correlation between the ranks of the coaches and the sports writers!

16 - 38


Test your learning…Test your learning…

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16 - 39


This completes Chapter 16This completes Chapter 16

16 - 1 copyright © 2004 by the mcgraw-hill companies, inc. all rights reserved

Documents

mcgrawhill companies

rank test

test of hypothesis

test statistics

sign test continued

p slide

kruskalwallis test

wilcoxon ranksum test