Hypothesis Test about the Difference Between Two Population Proportions
Using Independent Random Samples
H0:

Ha:
p1  p2  0
1. p1  p2  0
2. p1  p2  0
3. p1  p2  0

T.S.
pˆ 1  pˆ 2
z =
pˆ (1  pˆ )(
1
1
 )
n1 n2
where
pˆ 
x1  x2
n1  n2
R.R. For a given value of 


1. reject H0 if z  z

2. reject H0 if z  -z
3. reject H0 if z  z/2 or if z  -z/2

Assumptions:
1. Independent random samples.
ˆ1 (1 
2. n1 and n2 are sufficiently large ( n1 p
sampling distribution of (
pˆ1 )  10 and n2 pˆ 2 (1  pˆ 2 )  10 ) such that the
pˆ1  pˆ 2 ) is approximately normal.
Example A law student believes that the proportion of registered Republicans in
favor of additional tax incentives is greater than the proportion of registered
democrats in favor of such incentives. The student acquired independent random
samples of 200 republicans and 200 Democrats and found that 109 Republicans
and 86 Democrats in favor of additional tax incentives. Use the data to test Ho: p RpD=0 versus Ha: pR – pD > 0. Use =.05.
Hypothesis Test for the Difference Between the Two Population Proportions
Sample
1
2
X
109
86
N
200
200
p̂
0.545000
0.430000
Test for difference = 0 (vs > 0):
Z = 2.30
P-Value = 0.011
Example Two antibiotics are to be compared. Cultures from 387 out 450 patients
treated with antibiotic A showed antibacterial activity while cultures from 278 out
of 350 patients treated with antibiotic B showed antibacterial activity. Is there
sufficient evidence to conclude a difference in the proportion of active cultures for
the two antibiotics? Use =.05.