1

Chapter 12: Inference for Proportions

12.1 Inference for a Population Proportion

12.2 Comparing Two Proportions

2

Sampling Distribution of p-hat

n  From Chapter 9: n  p-hat is an unbiased estimator of p.

n  standard deviation of p-hat:

10nN that Provided*

)1(^

≥

−=npp

pσ

3

Figure 12.1, p. 687

4

Conditions for Inference about a Proportion (p. 687)

n  SRS n  N at least 10n n  For a significance test of H0:p=p0:

n  The sample size n is so large that both np0 and n(1-p0) are at least 10.

n  For a confidence interval: n  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.

5

Can we make inferences about a proportion?

n  Exercises 12.4 and 12.5, p. 689

6

Normal Sampling Distribution

n  If these conditions are met, the distribution of p-hat is approximately normal, and we can use the z-statistic:

nppppz

)1( 00

0

^

−−=

7

Inference for a Population Proportion

n  Confidence Interval:

n  Significance test of H0: p=p0:

nppzp )1(^^

*^ −±

nppppz

)1( 00

0

^

−−=

8

Practice

n  Exercise 12.7, p. 694

9

Homework

n  Read all of 12.1 (pp. 684-697)

n  Exercises:

n  12.14, 12.15, p. 698

10

Choosing a Sample Size (p. 695)

n  Our guess p* can be from a pilot study, or we could use the most conservative guess of p*=0.5.

n  Solve for n.

n  Example 12.9, p. 696.

mnppZ ≤− )1( **

*

11

Practice

n  Exercises: n  12.10, p. 696

n  12.8, p. 694

12

Homework

n  Reading, Section 12.2: n  pp. 702-713

13

12.2 Comparing Two Proportions

14

Conditions: Confidence Intervals for Comparing Two Proportions

n  SRS from each population

n  N>10n from each population

n  All of these are at least 5:

)1(

)1(

2

^

2

2

^

2

1

^

1

1

^

1

pn

pn

pn

pn

−

−

15

Calculating a Confidence Interval for Comparing Two Proportions (p. 704)

n  Two prop:

n  Remember the one-prop formula:

2

2

^

2

^

1

1

^

1

^

*

2

^

1

^ )1()1()(

nnz

pppppp−

+−

±−

nppzp )1(^^

*^ −±

16

Practice Problem

n  12.23, p. 706

17

Significance Tests for Comparing Two Proportions

n  Example 12.12, p. 707 n  H0: p1=p2 vs. Ha: p1<p2

n  “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^

nnXXp

++=

18

Significance Tests for Comparing Two Proportions

n  The test statistic is:

21

21^

nnXXp

++=

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−=

21

^^

^

2

^

1

11)1(nn

pp

ppz

n  Where,

19

Conditions: Significance Test for Comparing Two Proportions

n  SRS from each population

n  N>10n from each population

n  All of these are at least 5:

)1(

)1(

^

2

^

2

^

1

^

1

pn

pn

pn

pn

−

−

20

HW

n  Read pp. 707-711 n  Procedures for difference of two proportions.

n  12.25, p. 712

21

Practice

n  Problems:

n  12.36, p. 720

n  12.37, p. 720

n  12.41, p. 721

n  Chapter 12 test on Thursday

n  Formulas provided:

nppzp )1(^^

*^ −±

nppppz

)1( 00

0

^

−−=

2

2

^

2

^

1

1

^

1

^

*

2

^

1

^ )1()1()(

nnz

pppppp−

+−

±−⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−=

21

^^

^

2

^

1

11)1(nn

pp

ppz