Name
1 sample z-Interval
for Proportions
1 sample z-interval
for Means (σ known)
1 sample z-test for
proportions
1 sample z-test for
Means (σ known)
Mean of Sampling
Distribution
Standard
Deviation/Error of
Sampling
Distribution
p-value
p-value > 
p-value < 
Decision
about Ho
Conclusion
about Ha
AP STATISTICS: CHAPTER
22
Comparing Two Proportions - Notes
We will compare 2 population
proportions by looking at
independent random samples
from the 2 populations and
comparing the difference in the
sample proportions.
Confidence Interval for the difference in sample
ˆ1  pˆ 2
proportions ( p
):
Conditions that must be satisfied when constructing a
confidence interval for the difference in 2
population proportions:

1. Random
selection for both samples
2. n 1 pˆ 1  10 , n 1 (1  pˆ1 )  10, n 2 pˆ 2  10, n 2 (1  pˆ 2 )  10
(i.e., the counts of successes and failures in
each sample are at least 10—some books use 5)
3. Both populations must be at least 10 times
their corresponding sample sizes
4. Independent samples
ˆ1  pˆ 2:
Confidence Interval for p
( pˆ 1  pˆ 2 )  z

*
pˆ 1(1  pˆ 1 )
n
1

pˆ 2(1  pˆ 2 )
n
Standard Error
2
2 Sample z test for the difference of population
proportions:
Ho: p1 = p2
Ha: p1 (> < ≠) p2
Because we think p1 = p2, pool the estimates together
to get a better estimate of the true p.
x1  x2
ˆ
Pooled sample proportion: p 
n 1 n 2
(the pooled estimate of the common value of p1 and p2)
Conditions that must be satisfied when testing
hypotheses for the difference in 2 population
proportions:
1.
Random selection for both samples
2. n 1 pˆ  10 , n 1 (1  pˆ)  10, n 2 pˆ  10, n 2 (1  pˆ )  10
where ê is the pooled estimate of the common value of
p1 and p2 (some books use 5)
3. Both populations must be at least 10 times
their corresponding sample sizes
4.
Independent samples
Test statistic:
Standard error =
z 
pˆ1  pˆ2
 1
1 
ˆ
ˆ


p (1  p )

n2 
 n1
 1
1
pˆ(1  pˆ)

n2
 n1




(Notice that you are using the pooled in the standard
error formula)
On the formula sheet: use the
“unequal variances” formula for
standard error with confidence
intervals and the “equal variances”
formula for standard deviation
with significance tests.
Hypothesis testing: Difference of Proportions
In the past decade intensive antismoking campaigns
have been sponsored by both federal and private
agencies. Suppose the American Cancer Society
randomly sampled 1500 U.S. adults in 1985 and then
sampled 2000 U.S. adults in 1995 to determine
whether there was evidence that the percentage of
smokers had decreased. The results of the two sample
surveys are shown in the table below, where x1 and x2
represent the number of smokers in the 1985 and 1995
samples, respectively.
1985
n1 = 1500
1995
n2 = 2000
x1 = 576
x2 = 652
1. Construct a 95% confidence interval for the decrease in
the proportion of smokers in the U.S. from 1985 to 1995.
2.Do these data indicate that the proportion of U.S.
smokers decreased over this 10-year period? Use
 = .05.