Chapter 12
Chi Square Tests and Nonparametric
Tests
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-1

Learning Objectives
After completing this chapter, you should be able to:

Understand the

2 test for the difference between two proportions

Use a

2 test for differences in more than two proportions

Perform a

2 test of independence

Apply and interpret the Wilcoxon rank sum test for the difference between two medians

Perform and interpret the nonparametric Kruskal-Wallis rank test for the difference between three or more medians
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-2


2
H
0
: π
1
= π
2
H
1
: π
1
 π
2
The Chi-square test statistic is:
 2   all cells
( f o
 f e
) 2 f e



The

2 test produces the same results as the Z test for the difference in two proportions.
The

2 test can not be used for upper and lower tail tests only two-tail tests.
Therefore use the Z test for the difference in two proportions as it is a more complete test.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-3


2






Compares observed frequencies to expected frequencies if null hypothesis is true
The closer the observed frequencies are to the expected frequencies, the more likely the H
0 is true
Tests for equality (=) of proportions only
One variable with several groups or levels
Uses a contingency table


Assumptions:
 Independent random samples
“Large” sample size
All expected frequencies

5
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-4


2
 Extend the

2 test to the case with more than two independent populations:
H
0
: π
1
= π
2
= … = π c
H
1
: Not all of the π j are equal (j = 1, 2, …, c)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-5

The Chi-Square Test Statistic
The Chi-square test statistic is:

2   all cells
( f o
 f e f e
)
2

 where:
 f o
= observed frequency in a particular cell of the 2 x c table f e
= expected frequency in a particular cell if H
0 is true

2 for the 2 x c case has (2-1)(c-1) = c - 1 degrees of freedom
Assumed: each cell in the contingency table has expected frequency of at least 5
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-6


2 Test with More Than Two
Proportions: Example
 The sharing of patient records is a controversial issue in health care. A survey of 500 respondents asked whether they objected to their records being shared by insurance companies, by pharmacies, and by medical researchers. The results are summarized on the following table:
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-7

Object to
Record
Sharing
Yes

2 Test with More Than Two
Proportions: Example
Insurance
Companies
Organization
Pharmacies Medical
Researchers
410 295 335
1040
No 90 205 165
460
500 500 500
1500
Chap 12-8 Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.

Organization
Object to
Record
Sharing
Yes
Insurance
Companies
Pharmacies Medical
Researchers
295 335 1040
No
410
346.667
90
153.333
500
205
500
165
500
460
1500
Row Total /Grand Total x Column Total > 1040/1500x500 = 346.667
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-9


2 Test with More Than Two
Proportions: Example
Object to
Record
Sharing
Yes
No
Organization
Insurance
Companies f o
= 410 f e
= 346.667
f o
= 90 f e
= 153.333
Pharmacies f o
= 295 f e
= 346.667
f o
= 205 f e
= 153.333
Medical
Researchers f o
= 335 f e
= 346.667
f o
= 165 f e
= 153.333
All Expected Frequencies are > 5
The sample size is ‘Large’
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-10


2
(continued)
Compute Test Statistic: f
0 f e
410 346.667
295 346.667
(f
0
- f e
) 2 / f e
11.571
7.700
335 346.667
90 153.333
.393
26.159
205 153.333
17.409
165 153.333
.888
Test Statistic

2 = 64.120
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-11


2 Test with More Than Two
Proportions: Example
H
0
: π
1
= π
2
= π
3
H
1
: Not all of the π j are equal (j = 1, 2, 3)

2 (64.120) = 1.19E-14 is from the chi-square distribution with 2 degrees of freedom.
p -value = 1.19E-14
Conclusion: Since 1.19E-14 < .05, you reject H
0 and you conclude that at least one proportion of respondents who object to their records being shared is different from the other organizations
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-12

The Marascuilo Procedure
The Marascuilo procedure enables you to make comparisons between all pairs of groups (similar to Tukey-Kramer).
First, compute the observed differences p j among all c(c-1)/2 pairs.
- p j’
Second, compute the corresponding critical range for the Marascuilo procedure.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-13

The Marascuilo Procedure
 Critical Range for the Marascuilo Procedure:
Range j j
( ( 1 j j j j
) ) j j
/ /
( ( 1 j j
/ / j j
/ /
) )
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-14

The Marascuilo Procedure
Example
Organization
Object to
Record
Sharing
Insurance
Companies
Pharmacies Medical
Researchers
Yes 410 295 335
Proportion 410/500
P
1
= 0.82
295/500
P
2
= 0.59
335/500
P
3
= 0.67
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-15

The Marascuilo Procedure
Example
MARASCUILO TABLE
Proportions
| p Insurance – p Pharmacies |
| p Insurance – p Research |
| p Pharmacies – p Research |
Differences
.82 - .59 = .23
.82 - .67 = .15
.59 - .67 = -.08
Absolute
Differences
0.23
0.15
0.08
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-16

The Marascuilo Procedure
Critical Range
 
U
2 p j
( 1
 p j
)
 n j p j
/
( 1
 p j
/
) n j
/
Critical Range

5 .
991
.
82 ( 1

.
82 )
500

.
59 ( 1

.
59 )
500

.
0683
Critical Range

5 .
991
.
82 ( 1

.
82 )
500

.
67 ( 1

.
67 )
500

.
0665
Critical Range

5 .
991
.
59 ( 1

.
59 )
500

.
67 ( 1

.
67 )
500

.
0745
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-17

The Marascuilo Procedure
Example
Proportions
MARASCUILO TABLE
Absolute
Differences Critical Range
| p Insurance – p Pharmacies |
| p Insurance – p Research |
| p Research – p Insurance |
0.23
0.15
0.08
0.0683
0.0665
0.0745
Conclusion: Since each absolute difference is greater than the critical range, you conclude that each proportion is significantly different that the other two.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-18


2
 PhStat | Multiple-Sample Tests | Chi-Square
Test …
 Be sure to check the Marascuilo Procedure box to calculate which proportions are different (similar to the Tukey-Kramer Test)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-19


2 Test of Independence
 Similar to the

2 test for equality of more than two proportions, but extends the concept to contingency tables with r rows and c columns
H
0
: The two categorical variables are independent
(i.e., there is no relationship between them)
H
1
: The two categorical variables are dependent
(i.e., there is a relationship between them)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-20


2 Test of Independence
The Chi-square test statistic is:

 2   all cells
( f o
 f e
)
2 f e where: f o
= observed frequency in a particular cell of the r x c table f e
= expected frequency in a particular cell if H
0 is true

2 for the r x c case has (r-1)(c-1) degrees of freedom
Assumed: each cell in the contingency table has expected frequency of at least 1)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-21

Example: Test of Independence
 The meal plan selected by 200 students is shown below:
Class
Standing
Fresh.
Soph.
Junior
Senior
Total
Number of meals per week
20/week 10/week none
24
22
10
14
70
32
26
14
16
88
14
12
6
10
42
Total
70
60
30
40
200
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-22

Example: Test of Independence
 The hypothesis to be tested is:
H
0
: Meal plan and class standing are independent
(i.e., there is no relationship between them)
H
1
: Meal plan and class standing are dependent
(i.e., there is a relationship between them)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-23

(continued)
Class
Standing
Fresh.
Soph.
Junior
Senior
Total
Observed:
Number of meals per week
20/wk 10/wk none
24
22
10
14
70
32
26
14
16
88
14
12
6
10
42
Total
70
60
30
40
200
Class
Standing
Fresh.
Example for one cell:
Soph.
f e
 row total
 column total n
Junior
Senior

30
200

70

10 .
5
Total
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Expected cell frequencies if H
0 is true:
Number of meals per week
20/wk 10/wk none
24.5
30.8
14.7
21.0
26.4
12.6
10.5
13.2
6.3
14.0
17.6
70 88
8.4
42
Chap 12-24
Total
70
60
30
40
200

 The test statistic value is:

2 

 all cells
( f o
 f e
( 24

24 .
5 )
2 f e
)
2
24 .
5

( 32

30 .
8 )
2
30 .
8
  
( 10

8 .
4 )
2
8 .
4

0 .
7093 p -value=.9943
(continued)
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-25

The test statistic is

2 
0 .
7093 , p value

.9943
(continued)
Decision Rule:
If the p-value is <

, reject H
0
, otherwise, do not reject H
0
Here, p =value not <
 so do not reject H
0
Conclusion: There is not sufficient evidence that meal plan and class standing are related at

= .05
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-26


2
 PHStat | Multiple-Sample Tests | Chi-Square
Test …
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-27

McNemar Test


Used to test for the difference between two proportions of related samples (not independent)
You need to use a test statistic that follows the normal distribution
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-28

Condition 1
McNemar Test
Condition 2
Yes No
Yes
No
Totals
A
C
A+C
Totals
B
D
A+B
C+D
B+D A+B+C+D
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-29

McNemar Test Paired Data
Test Statistic
To test the hypothesis:
H
0
H
1
: π
: π
1
1
= π
2
H
0
: π
1
≠ π
2
H
1
: π
1
 π
2
H
0
: π
1
> π
2
H
1
: π
1
 π
2
< π
2
Use the test statistic:
Z

B

C
B

C
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-30

McNemar Test
Example
 Suppose you survey 300 homeowners and ask them if they are interested in refinancing their home. In an effort to generate business, a mortgage company improved their loan terms and reduced closing costs. The same homeowners were again surveyed. Determine if change in loan terms was effective in generating business for the mortgage company. The data are summarized as follows:
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-31

McNemar Test
Example
Survey response after change
Survey response before change
Yes No Totals
Yes 118 2 120
No 22 158 180
Totals 140 160 300
Test the hypothesis (at the 0.05 level of significance):
H
0
: π
1
≥ π
2
: The change in loan terms was ineffective
H
1
: π
1
< π
2
: The change in loan terms increased business
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-32

McNemar Test
Example
Survey response before change
Yes
No
Totals
Survey response after change
Yes No Totals
118
22
140
2
158
160
120
180
300
The test statistic is:
Z

B

C
B

C

2

22
2

22
 
4 .
08 p -value = 2.23E-05
Since 2.23E-05 < .05, you reject H
0 and conclude that the change in loan terms significantly increase business for the mortgage company.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-33

Wilcoxon Rank-Sum Test for
Differences in Two




Test two independent population medians
Populations need not be normally distributed
Distribution free procedure
Used when only rank data are available
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-34

Wilcoxon Rank-Sum Test
Using Normal Approximation
 For large samples, the test statistic T
1 with mean and standard deviation: is approximately normal
μ
T
1
 n
1
( n

1 )
2
σ
T
1
 n
1 n
2
( n
12

1 )



Must use the normal approximation if either n
1
Assign n
1 or n to be the smaller of the two sample sizes
2
> 10
Can also use the normal approximation for small samples n
1 and n
2
<10
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-35






Sample data are collected on the capacity rates (% of capacity) for two factories.
Are the median operating rates for two factories the same?
For factory A, the rates are 71, 82, 77, 94, 88
For factory B, the rates are 85, 82, 92, 97
Test for equality of the population medians at the 0.05 significance level
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-36

Wilcoxon Rank-Sum Test
Example with Small Samples
Ranked
Capacity values:
Tie in 3 rd and
4 th places
Capacity Rank
Factory A Factory B Factory A Factory B
71
77
82 82
85
1
2
3.5
3.5
5
88 6
92 7
94 8
97 9
Rank Sums:
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
20.5
24.5
Chap 12-37

Wilcoxon Rank-Sum Test
Example with Small Samples
Factory B has the smaller sample size, so the test statistic is the sum of the Factory B ranks:
T = 24.5
The sample sizes are: n
1
= 4 (factory B) n
2
= 5 (factory A)
The level of significance is α = .05
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-38

Wilcoxon Rank-Sum Test
Using Normal Approximation
 The Z test statistic is

T
1

T
1
T
1
 Where Z approximately follows a standardized normal distribution
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-39

Wilcoxon Rank-Sum Test
Using Normal Approximation
Use the setting of the prior example:
The sample sizes were: n
1
= 4 (factory B) n
2
= 5 (factory A)
The level of significance was α = .05
The test statistic was T
1
= 24.5
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-40

Wilcoxon Rank-Sum Test
Using Normal Approximation
μ
T
1
 n
1
( n

1 )
2

4 ( 9

1 )
2

20

σ
T
1
 n
1 n
2
( n

1 )
12

4 ( 5 ) ( 9

1 )

4 .
082
12
The test statistic is
Z

T
1
 μ
T
1
σ
T
1

24 .
5

20

1 .
10
4.082
 p -value =.2703 is not less than α = .05 so you do not reject H
0
– there is not sufficient evidence that the medians are different
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-41

 PHStat | Two-Sample Tests | Wilcoxon Rank
Sum Test …
 Use for both small samples <10 and large samples >10
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-42




Tests the equality of more than 2 population medians
Use when the normality assumption for one-way
ANOVA is violated
Assumptions:
 The samples are random and independent
 variables have a continuous distribution
 the data can be ranked
 populations have the same variability
 populations have the same shape
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-43

Kruskal-Wallis Rank Test
 The Kruskal-Wallis H test statistic:
(with c – 1 degrees of freedom)
H




12 n ( n

1 ) j c 

1
T j
2 n j




3 ( n

1 ) where: n = total number of values over the combined samples c = Number of groups
T j n j
= Sum of ranks in the j
= Size of the j th sample th sample
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-44

Kruskal-Wallis Rank Test
Example
 Do different branch offices have a different number of employees?
Office size
(Chicago, C)
23
41
54
78
66
Office size
(Denver, D)
55
60
72
45
70
Office size
(Houston, H)
30
40
18
34
44
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-45

Kruskal-Wallis Rank Test
Example
 Do different branch offices have a different number of employees?
Office size
(Chicago, C)
Ranking
23
41
54
78
66
2
6
9
15
12

= 44
Office size
(Denver, D)
55
60
72
45
70
Ranking
Office size
(Houston, H)
10
11
14
8
13

= 56
30
40
18
34
44
Ranking
3
5
1
4
7

= 20
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-46

Kruskal-Wallis Rank Test
Example
H
0
: Median
C

Median
D

Median
H
H
A
: Not all population Medians are equal
 The H statistic is
H





12 n ( n

1 ) j c 

1
T j
2 n j




3 ( n

1 )


 15
12
( 15

1 )


44
2
5

56
2
5

20
2
5






3 ( 15

1 )

6 .
72
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-47

Kruskal-Wallis Test
Example Solution
H
0
: Med
1
= Med
2
= Med
3
H
1
: Not all medians equal

= .05
H = 6.72
p -value = .0347
Reject at

= .05
There is sufficient evidence that the population median office sizes are not all equal by city.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-48

 PHStat | Multiple-Sample Tests | Kruskal-
Wallis Rank Test …
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-49

Chapter Summary
In this chapter, we have






Examined the

2 test for the difference between two proportions
Developed and applied the

2 test for differences in more than two proportions
Examined the

2 test for independence
Used the McNemar test for differences in two related proportions
Used the Wilcoxon rank sum test for two population medians
 Small Samples
 Large sample Z approximation
Applied the Kruskal-Wallis H-test for multiple population medians
Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.
Chap 12-50