Modern Business Statistics (6e)
Essentials of Modern
Business Statistics (7e)
Anderson, Sweeney, Williams, Camm, Cochran
© 2018 Cengage Learning
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
1
Modern Business Statistics (6e)
Chapter 12
Tests of Goodness of Fit, Independence,
and Multiple Proportions
 Goodness of Fit Test
 Test of Independence
 Testing For Equality of Three or More Population Proportions
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
2
Modern Business Statistics (6e)
Tests of Goodness of Fit, Independence,
and Multiple Proportions
 In this chapter we introduce three additional hypothesis-testing
procedures.
 The test statistic and the distribution used are based on the chisquare (χ²) distribution.
 In all cases, the data are categorical.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
3
Modern Business Statistics (6e)
Goodness of Fit Test:
Multinomial Probability Distribution, Part 1
1. State the null and alternative hypotheses.
H0: The population follows a multinomial distribution with
specified probabilities for each of the k categories
Ha: The population does not follow a multinomial distribution
with specified probabilities for each of the k categories
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
4
Modern Business Statistics (6e)
Goodness of Fit Test:
Multinomial Probability Distribution, Part 2
2. Select a random sample and record the observed frequency,
fi , for each of the k categories.
3. Assuming H0 is true, compute the expected frequency, e i , in
each category by multiplying the category probability by the
sample size.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
5
Modern Business Statistics (6e)
Goodness of Fit Test:
Multinomial Probability Distribution, Part 3
4. Compute the value of the test statistic.
where:
f i = observed frequency for category i
e i = expected frequency for category i
k = number of categories
Note: The test statistic has a chi-square distribution with k – 1 df
provided that the expected frequencies are 5 or more for
all categories.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
6
Modern Business Statistics (6e)
Goodness of Fit Test:
Multinomial Probability Distribution, Part 4
5. Rejection rule:
Where α is the significance level and there are k - 1 degrees of
freedom.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
7
Modern Business Statistics (6e)
Multinomial Distribution Goodness of Fit Test, Part 1
Example: Scott Marketing Research
Market share study conducted by Scott Marketing research has
identified that the market for a product X is shared by three
companies: A, B and C. Company C plans to introduce new and
improved product to replace its current entry in the market. It wants
Scott Marketing research to determine whether the new product will
alter the market share for the three companies.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
8
Modern Business Statistics (6e)
Multinomial Distribution Goodness of Fit Test, Part 2
Example: Scott Marketing Research
Using the historical market shares, we have a multinomial probability
distribution with p A = .30, p B = .50, p C = .20. Scott Marketing research
conducts a sample study using a consumer panel of 200 customers.
A hypothesis test can be used to determine whether the new product
of company C is likely to change the historical market shares for the
three companies. We will use an α = .05 level of significance.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
9
Modern Business Statistics (6e)
Multinomial Distribution Goodness of Fit Test, Part 3
Example: Scott Marketing Research
Hypotheses
H0: p A =.30, p B = .50, p C = .20
Ha: The probabilities are not p A =.30, p B = .50, p C = .20
where:
p A = probability a customer purchases the company A product
p B = probability a customer purchases the company B product
p C = probability a customer purchases the company C product
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
10
Modern Business Statistics (6e)
Multinomial Distribution Goodness of Fit Test, Part 4
Example: Scott Marketing Research
 Expected Frequencies
Category
Company A
Company B
Company C
Total
Expected Frequency
200 (.30) 60
200 (.50) 100
200 (.20) = 40
200
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
11
Modern Business Statistics (6e)
Multinomial Distribution Goodness of Fit Test, Part 5
Example: Scott Marketing Research
 Observed Frequencies (from the sample study)
Category
Company A
Company B
Company C
Total
Observed frequency
48
98
51
200
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
12
Modern Business Statistics (6e)
Multinomial Distribution Goodness of Fit Test, Part 6
Example: Scott Marketing Research
 Test Statistic
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
13
Modern Business Statistics (6e)
Multinomial Distribution Goodness of Fit Test, Part 7
Example: Scott Marketing Research
 Conclusion Using the p-Value Approach
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
14
Modern Business Statistics (6e)
Multinomial Distribution Goodness of Fit Test, Part 8
Example: Scott Marketing Research
 Conclusion Using the Critical Value Approach
With α = .05 and 2 degrees of freedom, the critical value for the
test statistic is χ² = 5.991.
Rejection rule
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
15
Modern Business Statistics (6e)
Multinomial Distribution Goodness of Fit Test, Part 9
Example: Scott Marketing Research
Bar chart of market shares by
company before and after the new
product for company C
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
16
Modern Business Statistics (6e)
Multinomial Distribution Goodness of Fit Test, Part 10
Example: Scott Marketing Research - Excel Worksheet
Customer
Preference
1
B
2
A
3
C
5
C
6
A
7
A
8
A
9
C
10
A
11
C
12
B
13
C
14
A
199
C
200
C
Categories
Obs. Frequency
A
48
B
98
C
54
Total 200
Empty cell
Hyp. Probability
Exp. Frequency
0.3
=D10*$E$7
0.5
=D11*$E$7
0.2
P-value
Hyp. Probability
Exp. Frequency
=D12*$E$7
0.3
60
=CHISQ.TEST(E4:E6,E10:E12)
0.5
100
0.2
40
P-value
0.0255
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
17
Modern Business Statistics (6e)
Test of Independence, Part 1
1. Set up the null and alternative hypotheses.
H0: The two categorical variables are independent
Ha: The two categorical variables are not independent
2. Select a random sample and record the observed frequency, fij , for each
cell of the contingency table.
3. Compute the expected frequency, eij , for each cell.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
18
Modern Business Statistics (6e)
Test of Independence, Part 2
4. Compute the test statistic.
5. Determine the rejection rule.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
19
Modern Business Statistics (6e)
Test of Independence, Part 3
Example: Beer Industry Association
Beer Industry Association conducts a survey to determine the
preferences of beer drinkers for light, regular and dark beers. It wants
to determine if preference for the beer is independent of the gender.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
20
Modern Business Statistics (6e)
Test of Independence, Part 4
Example: Beer Industry Association
A sample of 200 beer drinkers is taken. The sample result is
summarised in the following table.
 Observed frequencies
Beer Preference
Gender: Male
Gender: Female
Total
Light
51
39
90
Regular
56
21
77
Dark
25
8
33
Total
132
68
200
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
21
Modern Business Statistics (6e)
Test of Independence, Part 5
Example: Beer Industry Association
 Hypotheses
H0: Beer preference is independent of gender
Ha: Beer preference is not independent of gender
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
22
Modern Business Statistics (6e)
Test of Independence, Part 6
Example: Beer Industry Association
 Expected frequencies
Beer Preference
Gender: Male
Gender: Female
Total
Light
59.4
30.6
90
Regular
50.82
26.18
77
Dark
21.78
11.22
33
Total
132
68
200
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
23
Modern Business Statistics (6e)
Test of Independence, Part 7
Example: Beer Industry Association
Rejection Rule
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
24
Modern Business Statistics (6e)
Test of Independence, Part 8
Example: Beer Industry Association
 Computation of Chi-square test statistic
Beer
Preference
Gender
Observed
frequency
F subscript
Ij
Expected
frequency
E
subscript I
j
Light
Male
51
59.4
-8.4
70.56
1.19
Light
Female
39
30.6
8.4
70.56
2.31
Regular
Male
56
50.82
5.18
26.83
.53
Regular
Female
21
26.18
-5.18
26.83
1.02
Dark
Male
25
21.78
3.22
10.37
.48
Dark
Female
8
11.22
-3.22
10.37
.92
Empty cell
Total
200
200
Empty cell
Empty cell
Chi squared
equals 6.45
Difference
Left parenthesis f
subscript I j baseline
minus e subscript I j
baseline right
parenthesis
Squared difference
Left parenthesis f
subscript I j baseline
minus e subscript I j
baseline right
parenthesis squared
Squared
difference /
Expected
frequency
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
25
Modern Business Statistics (6e)
Test of Independence, Part 9
Example: Beer Industry Association
 Conclusion Using the p-Value Approach
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
26
Modern Business Statistics (6e)
Test of Independence, Part 10
Example: Beer Industry Association
 Conclusion Using the Critical Value Approach
We reject, at the .05 level of significance,
the assumption that the beer preference is
independent of the gender of the beer drinker.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
27
Modern Business Statistics (6e)
Test of Independence, Part 11
Example: Beer Industry
Association
 Bar Chart comparison
of Beer preference
by Gender
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
28
Modern Business Statistics (6e)
Test of Independence, Part 12
 Example: Beer Industry Association - Excel Worksheet
Beer Drinker
Preference
Gender
Count of Beer Drinker Preference
Gende
r Male
Female
Total
1
Regular
Male
Light
51
39
90
Regular
56
21
77
2
Light
3
Regular
Female
Dark
25
8
33
Male
Total
132
68
200
4
Regular
Male
5
Regular
Female
6
Regular
Male
7
Dark
Male
8
Dark
Male
9
Dark
Male
10
Light
Female
11
Light
Male
12
Dark
Female
13
Regular
Female
14
Regular
15
Light
16
Regular
Male
199
Light
Male
200
Light
Male
Preference
Gender Male
Gender Female
Light
=(H5*F$8)/H$8
=(H5*G$8)/H$8
Regular
=(H6*F$8)/H$8
=(H6*G$8)/H$8
Dark
=(H7*F$8)/H$8
=(H7*G$8)/H$8
Empty cell
Empty cell
=CHISQ.TEST(E4:E6,E10:E12)
Preference
Gender Male
Gender Female
Male
Light
59.40
30.60
Male
Regular
50.82
26.18
Dark
21.78
11.22
Empty cell
Empty cell
0.0398 (circled)
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
29
Modern Business Statistics (6e)
Testing for Equality of Population Proportions
for Three or More Populations, Part 1
Using the notation
p1 = population proportion for population 1
p2 = population proportion for population 2
p k = population proportion for population k
The hypotheses for the equality of population proportions for k > 3
populations are as follows:
H0: p1 = p2 = . . . = p k
Ha: Not all population proportions are equal
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
30
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 2
 If H0 cannot be rejected, we cannot detect a difference among the k
population proportions.
 If H0 can be rejected, we can conclude that not all k population
proportions are equal.
 Further analyses can be done to conclude which population
proportions are significantly different from others.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
31
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 3
Example: Customer loyalty for automobiles
Suppose in a particular study we want to compare the customer loyalty
for three automobiles: Chevrolet Impala, Ford Fusion and Honda
Accord.
p1 = population proportion for Chevrolet Impala
p2 = population proportion for Ford Fusion
p3 = population proportion for Honda Accord
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
32
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 4
Example: Customer loyalty for automobiles
 We begin by taking a sample of owners from each of the three
populations.
 Each sample contains categorical data indicating whether the
respondents are likely or not likely to repurchase the automobile.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
33
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 5
Example: Customer loyalty for automobiles
 Observed frequencies (Sample results)
Likely to
Repurchase
Chevrolet
Impala
Ford
Fusion
Honda Accord
Total
Yes
69
120
123
312
No
56
80
52
188
Total
125
200
175
500
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
34
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 6
 Next, we determine the expected frequencies under the assumption
H0 is correct
Expected Frequencies
under the Assumption H0 is True
If a significant difference exists between the observed and expected
frequencies, H0 can be rejected.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
35
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 7
Example: Customer loyalty for automobiles
 Expected frequencies (Computed)
Likely to
Repurchase
Chevrolet
Impala
Ford
Fusion
Honda Accord
Total
Yes
78
124.8
109.2
312
No
47
75.2
65.8
188
Total
125
200
175
500
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
36
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 8
 Next, compute the value of the chi-square test statistic.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
37
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 9
 Computation of the Chi-Square Test Statistic.
Likely to
repurchase
Automobile
owner
Observed frequency
F subscript I j
Expected
frequency
E subscript
Ij
Difference
Left parenthesis f
subscript I j baseline
minus e subscript I j
baseline right parenthesis
Squared difference
Left parenthesis f
subscript I j baseline
minus e subscript I j
baseline right
parenthesis squared
Yes
Impala
69
78.0
-9.0
81.00
1.04
Yes
Fusion
120
124.8
-4.8
23.04
0.18
Yes
Accord
123
109.2
13.8
190.44
1.74
No
Impala
56
47.0
9.0
81.00
1.72
No
Fusion
80
75.2
4.8
23.04
0.31
No
Accord
52
65.8
Negative 13.8
190.44
2.89
Empty cell
Empty cell
500
500
Empty cell
Empty cell
Chi squared equals 7.89
Squared difference /
Expected frequency
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
38
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 10
 Rejection Rule
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
39
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 11
Example: Customer loyalty for automobiles
 Conclusion Using the p-Value Approach
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
40
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 12
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
41
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 13
 Excel Worksheets
Owner
Automobile
Likely Repurchase
1
Chevrolet Impala
Yes
2
Chevrolet Impala
3
4
Count of Owner Likely Repurchase
Automobile Chevrolet Impala
Ford
Fusion
Honda Accord
Total
Yes
Yes
69
120
123
312
Chevrolet Impala
No
No
56
80
52
188
Chevrolet Impala
Yes
Total
125
200
175
500
5
Chevrolet Impala
Yes
6
Chevrolet Impala
Yes
7
Chevrolet Impala
Yes
Likely Repurchase
Automobile Chevrolet Impala
Automobile Ford Fusion
Automobile Honda Accord
=(15*F7)/17
=(15*G7)/17
=(15*H7)/17
8
Chevrolet Impala
Yes
Yes
9
Chevrolet Impala
No
No
=(16*F7)/17
=(16*G7)/17
=(16*H7)/17
10
Chevrolet Impala
Yes
Empty cell
P-value
=CHISQ.TEST(F5:H6,F12:H13)
Empty cell
11
Chevrolet Impala
Yes
12
Chevrolet Impala
No
Likely Repurchase
Automobile Chevrolet Impala
Automobile Ford Fusion
Automobile Honda Accord
13
Chevrolet Impala
Yes
14
Chevrolet Impala
No
Yes
78
124.8
109.2
47
75.2
65.8
P-value
0.0193
Empty cell
15
Chevrolet Impala
No
No
499
Chevrolet Impala
No
Empty cell
500
Chevrolet Impala
No
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
42
Modern Business Statistics (6e)
Testing the Equality of Population Proportions
for Three or More Populations, Part 14
 We have concluded that the population proportions for the three
populations of automobile owners are not equal.
 To identify where the differences between population proportions
exist, we will rely on a multiple comparisons procedure.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
43
Modern Business Statistics (6e)
Multiple Comparisons Procedure, Part 1
 We begin by computing the three sample proportions.
 We will use a multiple comparison procedure known as the
Marascuillo procedure.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
44
Modern Business Statistics (6e)
Multiple Comparisons Procedure, Part 2
 Marascuillo Procedure
We compute the absolute value of the pairwise difference between
sample proportions.
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
45
Modern Business Statistics (6e)
Multiple Comparisons Procedure, Part 3
 Critical Values for the Marascuillo Pairwise Comparison
For each pairwise comparison compute a critical value as follows:
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
46
Modern Business Statistics (6e)
Multiple Comparisons Procedure, Part 4
 Pairwise Comparison Tests
Pairwise Comparison
Absolute value of p overbar
subscript i minus p overbar
subscript j
C V subscript i j
Significant if absolute value
of p overbar subscript i minus
p overbar subscript j is
Greater than C V subscript i j
Chevrolet Impala and
Ford Fusion
.0480
.1380
Not significant
Chevrolet Impala and
Honda Accord
.1509
.1379
Significant
Ford Fusion and Honda
Accord
.1029
.1198
Not significant
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
47
Modern Business Statistics (6e)
End of Chapter 12
© 2018 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
48