Business statistics a decision making approach 6th edition ch16ppln

Chapter GoalsAfter completing this chapter, you should be able to:  Recognize when and how to use the Wilcoxon signed rank test for a population median  Recognize the situations for wh

Chapter Goals

After completing this chapter, you should be able to:

 Recognize when and how to use the Wilcoxon signed rank test for a population median

 Recognize the situations for which the Wilcoxon signed rank test applies and be able to use it for decision-making

 Know when and how to perform a Mann-Whitney U-test

 Perform nonparametric analysis of variance using the Kruskal-Wallis one-way ANOVA

Nonparametric Statistics

 Nonparametric Statistics

 Fewer restrictive assumptions about data levels and underlying probability distributions

 Population distributions may be skewed

 The level of data measurement may only

be ordinal or nominal

Wilcoxon Signed Rank Test

 Used to test a hypothesis about one

population median

 the median is the midpoint of the distribution: 50%

below, 50% above

 A hypothesized median is rejected if sample results vary too much from expectations

 no highly restrictive assumptions about the shape of the population distribution are needed

The W Test Statistic

Performing the Wilcoxon Signed Rank Test

Calculate the test statistic W using these steps:

 Step 1: collect sample data

 Step 2: compute d i = difference between each value and the hypothesized median

 Step 3: convert d i values to absolute differences

The W Test Statistic

Performing the Wilcoxon Signed Rank Test

 Step 4: determine the ranks for each d i value

 eliminate zero d i values

 Lowest d i value = 1

 For ties, assign each the average rank of the

tied observations

(continued)

The W Test Statistic

Performing the Wilcoxon Signed Rank Test

 Step 5: Create R+ and R- columns

 for data values greater than the hypothesized

median, put the rank in an R+ column

 for data values less than the hypothesized

median, put the rank in an R- column

(continued)

The W Test Statistic

Performing the Wilcoxon Signed Rank Test

 Step 6: the test statistic W is the sum of the ranks in the R+ column

 Test the hypothesis by comparing the calculated W to the critical value from the table in appendix P

 Note that n = the number of non-zero d i values

(continued)

 The median class size is claimed to be 40

 Sample data for 8 classes is randomly obtained

 Compare each value to the hypothesized median to find

difference

Class size = x i

Difference

d i = x i – 40 | d i | 23

45 34 78 34 66 61 95

-17 5 -6 38 -6 26 21 55

17 5 6 38 6 26 21 55

 Rank the absolute differences:

| d i | Rank

5 6 6 17 21 26 38 55

1 2.5 2.5 4 5 6 7 8

tied

(continued)

 Put ranks in R+ and R- columns and find sums:

Class size = x i

-17 5 -6 38 -6 26 21 55

17 5 6 38 6 26 21 55

4 1 2.5 7 2.5 6 5 8

1 7

6 5 8

4 2.5 2.5

Completing the Test

H 0 : Median = 40

H A : Median ≠ 40

Test at the α = 05 level:

This is a two-tailed test and n = 8, so find W L and W U in

appendix P: W L = 3 and W U = 33

The calculated test statistic is W = Σ R+ = 27

Completing the Test

H 0 : Median = 40

H A : Median ≠ 40

W L = 3 and W U = 33

W L < W < W U so do not reject H 0

(there is not sufficient evidence to conclude that the

median class size is different than 40)

If the Sample Size is Large

 The W test statistic approaches a normal distribution as n increases

 For n > 20, W can be approximated by

24

1) 1)(2n

n(n

4

1)

n(n W

z

+ +

+

−

=

where W = sum of the R+ ranks

d = number of non-zero d values

Nonparametric Tests for Two

Population Centers

Nonparametric Tests for Two Population Centers

Wilcoxon Matched-Pairs Signed Rank Test

Mann-Whitney

U-test

Large Samples

Small Samples

Large Samples Small

Samples

Mann-Whitney U-Test

Used to compare two samples from two populations

Assumptions:

 The two samples are independent and random

 The value measured is a continuous variable

 The measurement scale used is at least ordinal

 If they differ, the distributions of the two populations will

differ only with respect to the central location

 Consider two samples

 combine into a singe list, but keep track of which sample each value came from

 rank the values in the combined list from low

to high

 For ties, assign each the average rank of the tied values

 separate back into two samples, each value keeping its assigned ranking

 sum the rankings for each sample

Mann-Whitney U-Test

(continued)

 If the sum of rankings from one sample differs enough from the sum of rankings from the other sample, we conclude there is a difference in the population medians

Mann-Whitney U-Test

(continued)

2 1 1

2

1

R

) n

(

n n

n U

∑

−

+ +

2 1 2

2

1

R

) n

(

n n

n U

where:

n 1 and n 2 are the two sample sizes

∑ R 1 and ∑ R 2 = sum of ranks for samples 1 and 2

 Suppose the results are:

Class size (Math, M) Class size (English, E)

23 45 34 78 34 66 62 95 81

30 47 18 34 44 61 54 28 40

(continued)

Mann-Whitney U-Test

 Split back into the original samples:

Class size (Math, M) Rank

Class size (English, E) Rank

23 45 34 78 34 66 62 95 81

2 10 6 16 6 15 14 18 17

30 47 18 34 44 61 54 28 40

4 11 1 6 9 13 12 3 8

(continued)

Mann-Whitney U-Test

H 0 : Median M ≤ Median E

H A : Median M > Median E

Claim: Median class size for

Math is larger than the

median class size for English

22

104 2

(9)(10) (9)(9)

R 2

1) (n

n n

n

2 1

1 = + + − ∑ = + − =

59

67 2

(9)(10) (9)(9)

R 2

1) (n

n n

n

2 1

 The Mann-Whitney U tables in Appendices L and M give the lower tail of the U-distribution

 For one-tailed tests like this one, check the alternative hypothesis to see if U 1 or U 2 should be used as the test statistic

 Since the alternative hypothesis indicates that population 1 (Math) has a higher median, use U 1 as the test statistic

(continued)

Mann-Whitney U-Test

 Use U 1 as the test statistic: U = 22

 Compare U = 22 to the critical value Uα from the appropriate table

 For sample sizes less than 9, use Appendix L

 For samples sizes from 9 to 20, use Appendix M

 If U < Uα, reject H 0

(continued)

Mann-Whitney U-Test

Since U  U α , do not reject H 0

 Use U 1 as the test statistic: U = 19

 Uα from Appendix M for α = 05, n 1 = 9 and n 2 = 9 is Uα = 7

Mann-Whitney U-Test for

Large Samples

 The table in Appendix M includes Uα values only for sample sizes between 9 and 20

 The U statistic approaches a normal distribution as sample sizes increase

 If samples are larger than 20, a normal approximation can be used

Mann-Whitney U-Test for

12

) 1 n

n )(

n )(

n ( 1 2 1 + 2 +

= σ

Where n 1 and n 2 are sample sizes from populations 1 and 2

Mann-Whitney U-Test for

n )(

n )(

n (

2

n

n

U z

2 1

2 1

2 1

+ +

−

=

Large Sample Example

 We wish to test

 Suppose two samples are obtained:

 n 1 = 40 , n 2 = 50

 When rankings are completed, the sum of ranks for sample 1 is ΣR 1 = 1475

 When rankings are completed, the sum of ranks for sample 2 is ΣR 2 = 2620

H 0 : Median 1 ≥ Median 2

H A : Median 1 < Median 2

 U statistic is found to be U = 655

1345

1475 2

(40)(41) (40)(50)

R 2

1) (n

n n

n

2 1

1 = + + − ∑ = + − =

655

2620 2

(50)(51) (40)(50)

R 2

1) (n

n n

n

2 1

2 = + + − ∑ = + − =

Since the alternative hypothesis indicates that

population 2 has a higher median, use U 2 as the test

statistic

 Compute the U statistics:

Large Sample Example

(continued)

Since z = -2.80 < -1.645, we reject H 0

645 1

zα = −

Reject H 0

0 Median

Median

:

H

0 Median

Median

:

H

2 1

A

2 1

) 1 50 40

)(

50 )(

40 (

1000 655

12

) 1 n

n )(

n )(

n

(

2

n n U z

21

21

21

−

= +

+

−

= +

Wilcoxon Matched-Pairs

Signed Rank Test

 The Mann-Whitney U-Test is used when samples from two populations are independent

 If samples are paired, they are not independent

 Use Wilcoxon Matched-Pairs Signed Rank Test with paired samples

The Wilcoxon T Test Statistic

Performing the Small-Sample Wilcoxon

Matched Pairs Test (for n < 25)

Calculate the test statistic T using these steps:

 Step 1: collect sample data

 Step 2: compute d i = difference between the sample 1 value and its paired sample 2 value

 Step 3: rank the differences, and give each rank the same sign as the sign of the difference value

The Wilcoxon T Test Statistic

Performing the Small-Sample Wilcoxon

Matched Pairs Test (for n < 25)

 Step 4: The test statistic is the sum of the absolute values of the ranks for the group with the smaller expected sum

 Look at the alternative hypothesis to determine

the group with the smaller expected sum

 For two tailed tests, just choose the smaller sum

(continued)

Small Sample Example

 Paired samples, n = 9:

Value (before) Value (after)

38 45 34 58 30 46 42 55 41

30 47 18 34 34 31 24 38 40

b a

A

b a

0

Median Median

: H

Median Median

Small Sample Example

 Paired samples, n = 9:

Value (before) (after) Value

30 47 18 54 38 31 24 62 40

6 -2 16 4 -8 15 18 -7 1

4 -2 8 3 -6 7 9 -5 1

 The calculated T value is T = 13

 Complete the test by comparing the calculated T value to the critical T-value from Appendix N

 For n = 9 and α = 025 for a one-tailed test,

Small Sample Example

(continued)

Wilcoxon Matched Pairs Test

for Large Samples

 The table in Appendix N includes Tα values only for sample sizes from 6 to 25

 The T statistic approaches a normal distribution as sample size increases

 If the number of paired values is larger than 25, a normal approximation can be used

 The mean and standard deviation for Wilcoxon T :

(continued)

4

) 1 n

(

n +

= µ

24

) 1 n

2 )(

1 n

)(

n

= σ

where n is the number of paired values

Wilcoxon Matched Pairs Test

for Large Samples

Mann-Whitney U-Test for

2 )(

1 n

( n

4

) 1 n

(

n

T z

+ +

+

−

=

 Tests the equality of more than 2 population medians

 Assumptions:

 variables have a continuous distribution.

 the data are at least ordinal.

 samples are independent.

 samples come from populations whose only possible difference is that at least one may have a different central location than the others.

Kruskal-Wallis One-Way

ANOVA

Kruskal-Wallis Test Procedure

 Obtain relative rankings for each value

 In event of tie, each of the tied values gets the average rank

 Sum the rankings for data from each of the k groups

 Compute the H test statistic

Kruskal-Wallis Test Procedure

 The Kruskal-Wallis H test statistic:

(with k – 1 degrees of freedom)

) 1 N

(

3 n

R )

1 N

( N

R i = Sum of ranks in the i th sample

n i = Size of the i th sample

(continued)

 Complete the test by comparing the calculated H value to a critical χ2 value from the chi-square distribution with k – 1 degrees of freedom

(The chi-square distribution is Appendix G)

 Decision rule

 Reject H 0 if test statistic H > χ 2 α

 Otherwise do not reject H 0

(continued)

Kruskal-Wallis Test Procedure

 Do different departments have different class sizes?

Kruskal-Wallis Example

Class size (Math, M) (English, E) Class size (History, H) Class size

23 45 54 78 66

55 60 72 45 70

30 40 18 34 44

 Do different departments have different class sizes?

Kruskal-Wallis Example

Class size

(Math, M) Ranking

Class size (English, E) Ranking

Class size (History, H) Ranking

23 41 54 78 66

2 6 9 15 12

55 60 72 45 70

10 11 14 8 13

30 40 18 34 44

3 5 1 4 7

 The H statistic is

(continued)

Kruskal-Wallis Example

72 6 )

1 15

(

3 5

20 5

56 5

44 )

1 15

( 15

12

) 1 N

(

3 n

R )

1 N

( N

12 H

2 2

= +

=

+

− +

=

equal are

Medians population

all ot N : H

Median Median

Median :

H

A

H E

M

2 05

χ

 Compare H = 6.72 to the critical value from the chi-square distribution for 5 – 1 = 4 degrees of freedom and α = 05:

4877

9

2 05

χ

There is not sufficient evidence to reject that the population medians are all equal

Kruskal-Wallis Correction

 If tied rankings occur, give each observation the mean rank for which it is tied

 The H statistic is influenced by ties, and should be corrected

 Correction for tied rankings:

N N

) t t

(

1 3

g

1 i

i

3 i

g = Number of different groups of ties

N = Total number of observations

H Statistic Corrected for

Tied Rankings

 Corrected H statistic:

N N

) t t

( 1

) 1 N

(

3 n

R )

1 N

( N

12 H

3

g

1 i

i

3 i

k

1

i i

2 i

Chapter Summary

 Developed and applied the Wilcoxon signed rank test for a population median

W- Small Samples

 Large sample z approximation

 Developed and applied the Mann-Whitney U-test for two population medians

 Small Samples

 Large Sample z approximation

 Used the Wilcoxon Matched-Pairs T-test for paired

samples

 Small Samples

 Large sample z approximation

 Applied the Kruskal-Wallis H-test for multiple

population medians