### Ch 12 – Inference for Proportions YMS 12.1

```CH 12 – INFERENCE FOR PROPORTIONS
YMS 12.1 Inference for a Population Proportion
Ch 9 Sampling Distributions




p is an unbiased estimator of population proportion p
Standard deviation of is
if the population
p(1 p)
p
is at least 10 times n
n
Sampling distribution of is approximately normal if

np and n(1-p) are at least p10
Use z-scores to standardize
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
p in standard deviation formula
Test of Significance Ho: p = po
Verify that npo and n(1-po) are at least 10
p  po
 Formula
z
po (1  p o )
n


Confidence Interval
 and n(1- p) are at least 10
 Verify that n p

Form
p (1  p )
p  z *
n
Choosing the Sample Size

Margin of error
p *(1 p*)
z*
m
n

Use a guess for p*
Based on previous data
 Use the conservative estimate of 0.5 (unless you believe
is
closer to p0 or 1 because then p* = 0.5 will give you a much
larger sample size than necessary)

Which to use in formulas
and conditions?

Hypothesis Tests
 Use
po because that is the distribution you’re comparing

Confidence Intervals
 because you don’t have any other values
 Use p
(remember you’re using the CI to estimate the true
proportion p)
p698 #12.14-12.15, 12.17
YMS 12.2
Comparing Two Proportions
Sampling Distribution of

p1  p 2
When samples are large, the sampling distribution is
approximately normal.

Mean

Variance
 p  p   p   p  p1  p2
1
2

2
p1  p 2
1
2
 
2
p1
2
p 2
p1 (1  p1 ) p2 (1  p2 )


n1
n2
Confidence Intervals for Comparing Two
Proportions


Same form as for two means and standard error is
replacing p with
p
Conditions are still SRS, a population at least 10
times n, but now n1 , n1(1p1 ), n2 p1 , andpn2 2(1 2 greater than 5
) are pall
p706 #12.22-12.23
Pooled Sample Proportion


Because both samples actually come from one
huge population, we combine the sample results to
estimate the unknown population proportion p
Formula
X1  X 2
n1  n2
Significance Tests for Two Props



1and pwith
 standard error formula and
Replace p
pooled p
in
2
the conditions for count of successes and failures
Other conditions remain the same!
Test Stat
z
pˆ1  pˆ 2
1 1
pˆ (1  pˆ )   
 n1 n2 
p707 #12.24-12.26
A Civil Action – text, video and article
```