CH 12 – INFERENCE FOR PROPORTIONS

YMS 12.1Inference for a Population Proportion

Ch 9 Sampling Distributions

is an unbiased estimator of population proportion p

Standard deviation of is if the population is at least 10 times n

Sampling distribution of is approximately normal if np and n(1-p) are at least 10

Use z-scores to standardize

p

p

p

p p

n

( )1

Conditions for Inference

To be representative: Data are from an SRS from the population of interest

To accurately calculate standard deviation: Population is at least 10 times n

To use normal calculations: Counts of successes/failures must be at least 10

Standard Error Replacing p with in standard deviation

formula

Test of Significance Ho: p = po

Verify that npo and n(1-po) are at least 10 Formula

Confidence Interval Verify that n and n(1- ) are at least 10 Form

p p

p

zp p

p pn

o

o o

( )1

*( )

p zp p

n

1

Choosing the Sample Size

Margin of error

Use a guess for p* Based on previous data Use the conservative estimate of 0.5 (unless

you believe is closer to 0 or 1 because then p* = 0.5 will give you a much larger sample size than necessary)

p

zp p

nm*

* ( * )1

Which to use in formulasand conditions?

Hypothesis Tests Use po because that is the distribution

you’re comparing your result to Confidence Intervals

Use because you don’t have any other values (remember you’re using the CI to estimate the true proportion p)

p698 #12.14-12.15, 12.17

p

YMS 12.2Comparing Two Proportions

Sampling Distribution of When samples are large, the sampling

distribution is approximately normal.

Mean

Variance

p p1 2

p p p p p p1 2 1 2 1 2

( ) ( )p p p p

p p

n

p p

n1 2 1 2

2 2 2 1 1

1

2 2

2

1 1

Confidence Intervals for Comparing Two Proportions Same form as for two means and

standard error is replacing p with

Conditions are still SRS, a population at least 10 times n, but now n1 , n1(1- ), n2 , and n2(1- ) are all greater than 5

p706 #12.22-12.23

p

p1 p1p 2

p 2

Pooled Sample Proportion

Because both samples actually come from one huge population, we combine the sample results to estimate the unknown population proportion p

FormulaX X

n n1 2

1 2

Significance Tests for Two Props

Replace and with pooled in standard error formula and the conditions for count of successes and failures

Other conditions remain the same!

Test Stat

p707 #12.24-12.26

A Civil Action – text, video and article

p1 p 2 p

1 2

1 2

ˆ ˆ

1 1ˆ ˆ(1 )

p pz

p pn n