Business statistics a decision making approach 6th edition ch16ppln

Business Statistics:
A Decision-Making Approach
6th Edition

Chapter 16
Introduction to
Nonparametric Statistics

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-1


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

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-2


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


Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-3


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

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-4


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 di = difference between each value and the hypothesized median



Step 3: convert di values to absolute differences

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-5


The W Test Statistic
(continued)
Performing the Wilcoxon Signed Rank Test



Step 4: determine the ranks for each di value



eliminate zero di values



Lowest di value = 1



For ties, assign each the average rank of the
tied observations

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-6


The W Test Statistic
(continued)
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

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-7


The W Test Statistic
(continued)
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 di values

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-8


Example





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 Difference
size = xi di = xi – 40
23
45
34
78
34
66
61
95

-17
5
-6
38
-6
26
21
55

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

| di |
17
5
6
38
6
26
21

55
Chap 16-9


Example
(continued)


Rank the absolute differences:

tied

| di |

Rank

5
6
6
17
21
26
38
55

1
2.5
2.5
4
5

6
7
8

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-10


Example
(continued)


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

Class Difference
size = xi di = xi – 40
These three
are below
the claimed
median, the
others are
above

23
45
34
78
34

66
61
95

-17
5
-6
38
-6
26
21
55

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

| di |

Rank

17
5
6
38
6
26
21
55

4
1

2.5
7
2.5
6
5
8

R+

R4

1
2.5
7
2.5
6
5
8
Σ = 27

Σ=9

Chap 16-11


Completing the Test
H0: Median = 40
HA: Median ≠ 40

Test at the α = .05 level:

This is a two-tailed test and n = 8, so find WL and WU in
appendix P: WL = 3 and WU = 33
The calculated test statistic is W = ΣR+ = 27

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-12


Completing the Test
(continued)
H0: Median = 40
HA: Median ≠ 40

WL = 3 and WU = 33
W = ΣR+ = 27
reject H0

WL = 3

do not reject H0

WL < W < WU

WU = 33

reject H0

so do not reject H0

(there is not sufficient evidence to conclude that the
median class size is different than 40)
Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-13


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

z=

n(n + 1)
W−
4
n(n + 1)(2n + 1)
24

where W = sum of the R+ ranks
d = number of non-zero di values
Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-14

Nonparametric Tests for Two
Population Centers
Nonparametric
Tests for Two
Population Centers

Mann-Whitney
U-test
Small
Samples

Large
Samples

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Wilcoxon
Matched-Pairs
Signed Rank Test
Small
Samples

Large
Samples
Chap 16-15


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

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-16


Mann-Whitney U-Test
(continued)


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

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-17


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

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-18


Mann-Whitney U-Test
(continued)

Mann-Whitney U-Statistics

n1 (n1 + 1)
U1 = n1n 2 +
− ∑ R1
2
n2 (n2 + 1)
U2 = n1n2 +
− ∑ R2
2
where:
n1 and n2 are the two sample sizes
∑R1 and ∑R2 = sum of ranks for samples 1 and 2
Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-19


Mann-Whitney U-Test
(continued)

Claim: Median class size for Math is larger
than the median class size for English
A random sample of 9 Math and 9 English
classes is selected (samples do not have to
be of equal size)
Rank the combined values and then split
them back into the separate samples
Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-20

Mann-Whitney U-Test
(continued)


Suppose the results are:

Class size (Math, M) Class size (English, E)
23
45
34
78
34
66
62
95
81
Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

30
47
18
34
44
61
54
28
40
Chap 16-21

Mann-Whitney U-Test
(continued)

Ranking for combined samples

tied

Size

Rank

Size

Rank

18

1

45

10

23

2

47

11

28

3

54

12

30

4

61

13

34

6

62

14

34

6

66

15

34

6

78

16

40

8

81

17

44

9

95

18

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-22


Mann-Whitney U-Test


(continued)

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

Σ = 104

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Σ = 67
Chap 16-23


Mann-Whitney U-Test
(continued)
Claim: Median class size for
Math is larger than the
median class size for English

Math:

H0: MedianM ≤ MedianE
HA: MedianM > MedianE

n1(n1 + 1)
(9)(10)
U1 = n1n2 +
− ∑ R1 = (9)(9) +
− 104 = 22
2
2

n2 (n2 + 1)
(9)(10)
− ∑ R 2 = (9)(9) +

− 67 = 59
English: U2 = n1n2 +
2
2
Note: U1 + U2 = n1n2

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-24


Mann-Whitney U-Test
(continued)


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 U2 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

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 16-25