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Chapter 12: Inference for Proportions
12.1 Inference for a Population Proportion
12.2 Comparing Two Proportions
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Sampling Distribution of p-hat
From Chapter 9: p-hat is an unbiased estimator of p. standard deviation of p-hat:
10nN that Provided*
)1(^
n
ppp
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Figure 12.1, p. 687
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Conditions for Inference abouta Proportion (p. 687)
SRS N at least 10n For a significance test of H0:p=p0:
The sample size n is so large that both np0 and n(1-p0) are at least 10.
For a confidence interval: n is so large that both the count of successes, n*p-
hat, and the count of failures, n(1 - p-hat), are at least 10.
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Can we make inferences about a proportion?
Exercises 12.4 and 12.5, p. 689
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Normal Sampling Distribution
If these conditions are met, the distribution of p-hat is approximately normal, and we can use the z-statistic:
npp
ppz
)1( 00
0
^
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Inference for a Population Proportion
Confidence Interval:
Significance test of H0: p=p0:
n
ppzp
)1(^^
*^
npp
ppz
)1( 00
0
^
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Practice
Exercise 12.7, p. 694
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Homework
Read all of 12.1 (pp. 684-697) Exercises:
12.14, 12.15, p. 698
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Choosing a Sample Size (p. 695)
Our guess p* can be from a pilot study, or we could use the most conservative guess of p*=0.5.
Solve for n. Example 12.9, p. 696.
mn
ppZ
)1( ***
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Practice
Exercises: 12.10, p. 696 12.8, p. 694
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Homework
Reading, Section 12.2: pp. 702-713
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12.2 Comparing Two Proportions
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Conditions: Confidence Intervals for Comparing Two Proportions
SRS from each population N>10n from each population All of these are at least 5:
)1(
)1(
2
^
2
2
^
2
1
^
1
1
^
1
pn
pn
pn
pn
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Calculating a Confidence Interval for Comparing Two Proportions (p. 704)
Two prop:
Remember the one-prop formula:
2
2
^
2
^
1
1
^
1
^
*
2
^
1
^ )1()1()(
nnz
pppppp
n
ppzp
)1(^^
*^
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Practice Problem
12.23, p. 706
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Significance Tests forComparing Two Proportions
Example 12.12, p. 707 H0: p1=p2 vs. Ha: p1<p2
“If H0 is true, all observations in both samples really come from a single population of men of whom a single unknown proportion p will have a heart attack in a five-year period. So instead of estimating p1 and p2 separately, we pool the two samples and use the overall sample proportion to estimate the single population parameter p.
21
21^
nn
XXp
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Significance Tests forComparing Two Proportions
The test statistic is:
21
21^
nn
XXp
21
^^
^
2
^
1
11)1(
nnpp
ppz
Where,
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Conditions: Significance Test for Comparing Two Proportions
SRS from each population N>10n from each population All of these are at least 5:
)1(
)1(
^
2
^
2
^
1
^
1
pn
pn
pn
pn
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Practice Problem
12.25, p. 712
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Practice
Problems: 12.36, p. 720 12.37, p. 720 12.41, p. 721
Chapter 12 test on Monday Formulas provided:
n
ppzp
)1(^^
*^
npp
ppz
)1( 00
0
^
2
2
^
2
^
1
1
^
1
^
*
2
^
1
^ )1()1()(
nnz
pppppp
21
^^
^
2
^
1
11)1(
nnpp
ppz
