### Math 225 Course Notes

### Section 7.6
Hypothesis Testing: The Difference Between Two Population Proportions

Hypothesis testing for the difference in proportions
is based on the sampling distribution for the difference in sample proportions.
The methodology here is valid whenever both samples are sufficiently large
for their individual sampling distributions to be approximately normal.

We are interested in the different population rates
of hyperactivity
in two separate populations of retarded children.
Random samples of size 100 and 120 are respectively chosen,
with sample proportions of .38 and .40 respectively.
May we conclude that there is a difference
in the population proportions of hyperactive children
in these two populations?
The sampling distribution
for the difference in sample proportions
will be approximately normal
since each sample is sufficiently large.
(There are at least 5 children of each type in each sample.)

We state our hypotheses symbolically as

H_{0}: p_{1} = p_{2}
H_{A}: p_{1} does not equal p_{2}

Under the null hypothesis,
each population has a common proportion.
Instead of estimating the SE for the difference in sample proportions
by plugging the estimates in individually,
we can do (a little) better by pooling the information from
both samples to estimate the common p.
p = (x_{1} + x_{2}) / (n_{1} + n_{2})

or
p = (38 + 48) / (100 + 120) = 0.39

We may estimate the standard error by plugging in this estimate for p
for both populations,
finding
= 0.0660
The z statistic is

z = (.38 - .40) / 0.0660 = -0.30

The two-sided p-value is twice the area to the left of -0.30, or
2(.3821) = .7642.
There is no evidence against the null hypothesis,
so we do not reject it.

Last modified: April 15, 1996

Bret Larget,
larget@mathcs.duq.edu