Statistical techniques in business ecohomics chap016

Chapter Sixteen Nonparametric Methods: Analysis of Ranked Data GOALS When you have completed this chapter, you will be able to: ONE Conduct the sign test for dependent samples using th

Chapter Sixteen

Nonparametric Methods: Analysis of

Ranked Data

GOALS

When you have completed this chapter, you

will be able to:

ONE

Conduct the sign test for dependent samples using the binomial and standard normal distributions as the test statistics.

TWO

Conduct a test of hypothesis for dependent samples using the

Wilcoxon signed-rank test.

Chapter Sixteen

GOALS

When you have completed this chapter, you

will be able to: FIVE

Compute and interpret Spearman’s coefficient of rank correlation.

The test requires

dependent

(related) samples.

The Sign Test

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

samples and the

standard normal (z) for

large samples.

The Sign Test continued

n is the number of usable

pairs (without ties), X is

the number of pluses or minuses, and the binomial

If both n  and n(1-  ) are greater than 5,

the z distribution is appropriate

less than n/2, then

Example 1

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.

percent for one company

Step 4: H0 is rejected We conclude that R&D expense as a percent of income declined from 2000 to 2001

Testing a Hypothesis About a Median

Testing a Hypothesis About a

Median

When testing the value of

the median, we use the

Example 2

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

Example 2 Continued

450

$ median

: H

$450

= median

2 300

5

) 300 (

50 )

5 170 (

5

50 )

5 (

Wilcoxon Signed-Rank Test

The observations must

be related or dependent.

Use if assumption of

normality is violated

for the paired-t test

Wilcoxon Signed-Rank Test

Requires the ordinal scale of measurement

Wilcoxon Signed-Rank Test

Compare the smaller

of the two rank sums

with the T value,

obtained from Appendix H.

Wilcoxon Signed-Rank Test

Compute the

differences between

related observations.

Rank the absolute differences from low to high.

Return the signs to

the ranks and sum

positive and negative

ranks.

Example 3

Step 2

H0 is rejected if the smaller of the

rank sums is less than or equal to

5 See Appendix H

From Example 1

Have R&D expenses declined as a percent of income?

Use 05 significance level.

Step 1: H0 : The percent stayed the same.

H1 : The percent declined.

Example 3 Continued

The smaller rank sum is 5, which is equal to the critical

value of T H0 is rejected The

percent has declined from one

year to the next

Company 2000 2001 Difference ABS-Diff Rank R+

Wilcoxon Rank-Sum Test

Each sample must

contain at least

eight observations

Wilcoxon Rank-Sum Test

No assumption about the shape

of the population

is required

The data must

be at least ordinal scale

Wilcoxon Rank-Sum Test

Rank all data values

from low to high as if

they were from a

single population

Determine the sum of ranks for each of the two samples

Use the smaller of

the two sums W to

compute the test

) 1 (

2

) 1 (

2 1

2 1

2 1

n n

n n

n W

z

Wilcoxon Rank-Sum Test

Example 4

Hills Community College purchased two vehicles, a Ford and a

Chevy, for the administration’s use when traveling 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?

Example 4

914

0

12

) 1 9

8 )(

9 ( 8

2

) 1 9

8 (

8 5

81

12

) 1 (

2

) 1 (

2 1

2 1

2 1

n n

n n

n W

H1 : The populations are not the same.

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

Step 4: 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.

Kruskal-Wallis Test: Analysis of

Variance by Ranks

Kruskal-Wallis Test Analysis of Variance by 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

Kruskal-Wallis Test: Analysis of Variance by Ranks continued

Test Statistic

H

n n

R n

R n

R

k k

1

2 2

Example 5

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?

.

5

) 1 20 (

3 5

74 5

62 5

35 5

39 )

1 20 (

20

12

) 1 (

3

) (

) (

) (

)

( ) 1 (

12

2 2

2 2

2 4 2

2 3 2

2 2 1

2 1

R n

R n

R n

H1: The populations are not the same.

Step 2: H0 is rejected if  is greater than 7.185 There are 3 degrees of freedom at the 05 significance level

The null hypothesis is not rejected.

Rank-Order Correlation

Spearman’s Coefficient of Rank Correlation

Reports the association between

two sets of ranked

observations

Similar to Pearson’s

coefficient of correlation, but is based on ranked data

Ranges from –1.00 up to 1.00



d is the difference in the

ranks and n is the

number of observations

Testing the Significance of r s

not 0.

Ho: Rank

correlation in

population is 0

for the Atlantic Coast Conference

by the coaches and sports writers

Example 6 Continued

905

0 )

1 8

( 8

) 8 (

6 1

) 1 (

6 1

2

2 2

d

rs

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

Coefficient

of Rank

Correlation