Ch 12 – Inference for Proportions YMS 12.1
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Transcript 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 p0 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
your result to
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
p1 p 2
When samples are large, the sampling distribution is
approximately normal.
Mean
Variance
p p p p p1 p2
1
2
2
p1 p 2
1
2
2
p1
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(1p1 ), n2 p1 , andpn2 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 pwith
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
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