Essentials of Modern Business Statistics (7e)
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.
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Essentials of Modern Business Statistics (7e)
Chapter 10, Part B
Inference About Means and Proportions with Two Populations
 Inferences About the Difference Between Two 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.
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Essentials of Modern Business Statistics (7e)
Inferences About the Difference Between
Two Population Proportions
 Interval Estimation of p1 - p2
 Hypothesis Tests about p1 - p2
© 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.
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Essentials of Modern Business Statistics (7e)
Sampling Distribution of 𝑝1 − 𝑝2
 Expected Value
𝐸 𝑝1 − 𝑝2 = 𝑝1 − 𝑝2
 Standard Deviation (Standard Error)
𝜎𝑝1−𝑝2 =
𝑝1 (1 − 𝑝1 ) 𝑝2 (1 − 𝑝2 )
+
𝑛1
𝑛2
where: n1 = size of sample taken from population 1
n2 = size of sample taken from population 2
© 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.
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Essentials of Modern Business Statistics (7e)
Sampling Distribution of 𝑝1 − 𝑝2
 If the sample sizes are large, the sampling distribution of 𝑝1 − 𝑝2
can be approximated by a normal probability distribution.
 The sample sizes are sufficiently large if all of these conditions are
met:
n1p1 > 5 and n1(1 - p1) > 5
n2p2 > 5 and n2(1 - p2) > 5
© 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.
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Essentials of Modern Business Statistics (7e)
Interval Estimation of p1 - p2
 Interval Estimate
𝑝1 − 𝑝2 ± 𝑧𝛼/2
𝑝1 (1 − 𝑝1 ) 𝑝2 (1 − 𝑝2 )
+
𝑛1
𝑛2
© 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.
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Essentials of Modern Business Statistics (7e)
Interval Estimation of p1 - p2
Example: Tax preparation firm
A tax preparation firm is interested in comparing the quality of work at
two of its regional offices. By randomly selecting samples of tax returns
prepared at each office and verifying the sample return’s accuracy, the
firm will be able to estimate the proportion of erroneous returns
prepared at each office.
© 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.
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Essentials of Modern Business Statistics (7e)
Interval Estimation of p1 - p2
Example: Tax preparation firm
Independent random samples from the two offices provide the
following information:
Office 1
n1 = 250
Number of returns
with errors = 35
Office 2
n1 = 250
Number of returns
with errors = 27
© 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.
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Essentials of Modern Business Statistics (7e)
Point Estimator of the Difference Between
Two Population Proportions
Example: Tax preparation firm
p1 = proportion of erroneous returns for population 1 (office 1)
p2 = proportion of erroneous returns for population 2 (office 2)
𝑝1 = sample proportion for a random sample from population 1
𝑝2 = sample proportion for a random sample from population 1
35
27
𝑝1 − 𝑝2 =
−
= .14 − .09 = .05
250 300
© 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.
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Essentials of Modern Business Statistics (7e)
Interval Estimation of p1 - p2
Example: Tax preparation firm
For  = .1, z.05 = 1.645
.14(1 − .14) .09(1 − .09)
. 14 − .09 ± 1.645
+
250
300
.05 ± .045
The 90% confidence interval for the difference in the proportion of
erroneous tax returns prepared by two offices is .005 to .095.
© 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.
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Essentials of Modern Business Statistics (7e)
Interval Estimation of p1 - p2
 Excel Formula
and Value
worksheet
© 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.
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Essentials of Modern Business Statistics (7e)
Hypothesis Tests about p1 - p2
 We focus on tests involving no difference between the two
population proportions (i.e. p1 = p2)
H0: p1 – p2 > 0
Ha: p1 – p2 < 0
H0: p1 – p2 < 0
Ha: p1 – p2 > 0
Left-tailed
Right-tailed
H0: p1 – p2 = 0
Ha: p1 – p2 ≠ 0
Two-tailed
© 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.
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Essentials of Modern Business Statistics (7e)
Hypothesis Tests about p1 - p2
Standard Error of 𝑝1 − 𝑝2 when p1 = p2 = p
𝜎𝑝1−𝑝2 = 𝑝(1 − 𝑝)
1
𝑛1
+
1
𝑛2
Pooled Estimator of p when p1 = p2 = p
𝑛1 𝑝1 + 𝑛2 𝑝2
𝑝=
𝑛1 + 𝑛2
© 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.
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Essentials of Modern Business Statistics (7e)
Hypothesis Tests about p1 - p2
 Test Statistic
𝑧=
𝑝1 − 𝑝2
1
1
𝑝(1 − 𝑝)
+
𝑛1 𝑛2
© 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.
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Essentials of Modern Business Statistics (7e)
Hypothesis Tests about p1 - p2
Example: Tax preparation firm
Let us assume that the firm wants to use a hypothesis test to
determine whether the error proportions differ between the two
offices.
Can we conclude, using a .10 level of significance that the error
proportions differ between the two offices?
© 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.
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Essentials of Modern Business Statistics (7e)
Hypothesis Tests about p1 - p2
Example: Tax preparation firm
 p -Value approach
1. Develop the hypotheses.
H0: p1 - p2 = 0
Ha: p1 - p2 ≠ 0
p1 = proportion of erroneous returns for population 1 (office 1)
p2 = proportion of erroneous returns for population 2 (office 2)
© 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.
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Essentials of Modern Business Statistics (7e)
Hypothesis Tests about p1 - p2
Example: Tax preparation firm
 p -Value approach
2. Specify the level of significance.  = .10
3. Compute the value of the test statistic.
𝑝=
𝑠𝑝1−𝑝2 =
250 .14 +300(.09)
250+300
.1127(1 − .1127)
1
250
+
= .1127
1
300
= .0271
.14 − .09
.05
𝑧=
=
= 1.85
.0271
.0271
© 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.
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Essentials of Modern Business Statistics (7e)
Hypothesis Tests about p1 - p2
Example: Tax preparation firm
 p –Value Approach
4. Compute the p –value.
For z = 1.85, the area in the upper tail is 1.0000 - .9678 = .0322
Doubling this for a two-tailed test, p–value = 2(0.0322)
the p–value = .0644
5. Determine whether to reject H0.
Because p–value <  = .10, we reject H0.
We can conclude that the error rates differ between two offices.
© 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.
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Essentials of Modern Business Statistics (7e)
Hypothesis Tests about p1 - p2
 Excel
Formula
and Value
worksheet
© 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.
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Essentials of Modern Business Statistics (7e)
End of Chapter 10, Part B
© 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.
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