Statistics 8.4
8.4 Testing the Difference
Between Proportions
Statistics
Mrs. Spitz
Spring 2009
Statistics 8.4
Objectives/Assignment
• How to perform a z-test for the
difference between two population
proportions p1 and p2.
Assignment: pp. 404-406 #1-12
Statistics 8.4
Statistics 8.4
Two Sample z-Test for the Difference
Between Proportions
• In this section, you will learn how to use a ztest to test the difference between two
population proportions p1 and p2 using a
sample proportion from each population. For
instance, suppose you want to determine
whether the proportion of female college
students who earn a bachelor’s degree in
four years is different from the proportion of
male college students who earn a bachelor’s
degree in four years.
Statistics 8.4
Two Sample z-Test for the Difference
Between Proportions
• To use a z-test to test such a difference, the
following conditions are necessary:
1. The samples must be independent.
2. The samples must be large enough to use a
normal sampling distribution. That is:
n1p1 ≥ 5,
n1q1 ≥ 5,
n2p2 ≥ 5,
n2 q 2 ≥ 5
Statistics 8.4
Two Sample z-Test for the Difference
Between Proportions
• If these conditions are met, then the
sampling distribution for pˆ1  pˆ 2 , the
difference between the sample
proportions, is a normal distribution with
mean:
 pˆ  pˆ  p1  p2
1
2
And a standard error
 pˆ  pˆ 
1
2
p1q1 p2 q2

n1
n2
Statistics 8.4
Two sample z-test for difference
between proportions
• Notice that you need to know the
population proportions to calculate the
standard error. Because a hypothesis
test for p1 - p2 is based on the condition
of equality that p1 = p2 , you can
calculate a weighted estimate of p
using:
where
x1  x2
x

n
p̂
x

n
p̂
and
1
1
1
2
2
2
p
n1  n2
Statistics 8.4
Two-Sample z-Test for the Difference
Between Proportions
• Using the weighted estimate p , the
standard error of the sampling
distribution for pˆ1  pˆ 2 , is:
 pˆ  pˆ 
1
2
1 1
pq (  )
n1 n2
where
q  1 p
Also, when determining whether the z-test can be used
for the difference between proportions, you should use p
in place of p1 and p2 and q in place of q1 and q2
Statistics 8.4
Two Sample z-Test for the
Difference Between Proportions
ˆ1  pˆ 2 , is normal,
If the sampling distribution for p
you can use a two-sample z-test to test the difference
between two population proportions, p1 and p2
Statistics 8.4
If the null hypothesis states that p1 = p2,
then the expression, p1 - p2 is equal to 0 in
the preceding test.
Statistics 8.4
Ex. 1: A Two-Sample z-Test for the
Difference Between Proportions
• In a study of 200 adult female and 250
adult male Internet users, 30% of the
females and 38% of the males said that
they plan to shop online at least once
during the next month. At  = 0.10, test
the claim that there is a difference in
the proportion of female Internet users
who plan to shop online and the
proportion of male Internet users who
plan to shop online.
Statistics 8.4
Solution
• You want to determine whether there is a
difference in the proportions. So, the null and
alternative hypotheses are:
Ho: p1 = p2
and p1  p2 (claim)
Because the test is two-tailed and the level of
significance is  = 0.10, the critical values are
1.645. The rejection regions are z < -1.645
and z > 1.645.
Statistics 8.4
Solution
• The weighted estimate of the population
proportion is:
x1  x2
p
n1  n2
60  95
155
p

 0.344
200  250 450
• And q  1  p  1  0.344  0.656
. .
Because 200(0.344), 200(0.656),
250(0.344) and 250)0.656 are at least 5,
you can use the two-sample z-test.
Statistics 8.4
Solution
• The standardized test statistic is:
( pˆ 1  pˆ 2 )  ( p1  p2 )
z
1 1
pq (  )
n1 n2
(0.30  0.38)  (0)
z
 1.775
1
1
(0.344)(0.656)(

)
200 250
Statistics 8.4
Solution
• The graph shows the
location of the
rejection regions and
the standardized test
statistic. Because z
is in the rejection
region, you should
decide to reject the
null hypothesis. You
have enough
evidence at the 10%
level to conclude
there is a difference
in the proportion of
female and the
proportion of male
Internet users who
plan to shop online.
Statistics 8.4
Ex. 2: Two-Sample z-Test for the Difference
Between Proportions
• A medical research team conducted a study
to test the effect of a cholesterol-reducing
medication. At the end of the study, the
researchers found that of the 4700 subjects
who took the medication, 301 died of heart
disease. Of the 4300 subjects who took a
placebo, 357 died of heart disease. At  =
0.01, can you conclude that the death rate is
lower for those who took the medication than
for those who took the placebo?
Statistics 8.4
Solution
• You want to determine whether there is a
difference in the proportions. So, the null and
alternative hypotheses are:
Ho: p1 ≥ p2
and p1 < p2 (claim)
Because the test is left-tailed and the level of
significance is  = 0.01, the critical value is
-2.33. The rejection region is z < -2.33
Statistics 8.4
Solution
• The weighted estimate of the population
proportion is:
x1  x2
p
n1  n2
301  357
658
p

 0.073
4700  4300 9000
• And q  1  p  1  0.073  0.927
. .
Because 4700(0.073), 4700(0.927),
4300(0.073) and 4300)0.927 are at least
5, you can use the two-sample z-test.
Statistics 8.4
Solution
• The standardized test statistic is:
( pˆ 1  pˆ 2 )  ( p1  p2 )
z
1 1
pq (  )
n1 n2
(0.064  0.083)  (0)
z
 3.461
1
1
(0.073)(0.927)(

)
4700 4300
Statistics 8.4
Solution
• The graph shows the
location of the
rejection region and
the standardized test
statistic. Because z
is in the rejection
region, you should
decide to reject the
null hypothesis. At
the 1% level, there is
enough evidence to
conclude that the
death rate is lower for
those who took the
medication than for
those who took the
placebo.