### Math 225 Course Notes

### Section 5.6:
Distribution of the Difference Between Two Sample Proportions

This sampling distribution should be used whenever the situation
involves two independent samples from different populations
and the sample proportions are compared.
As in all examples presented in this chapter,
the
central limit theorem
allows the use of the
normal distribution
to find probabilities.
In this situation,
there are different formula
for the mean and standard error,
but the same logic and procedure for solving problems
remains the same.

Demonstration of these ideas is shown in an
example.

The sampling distribution of
is summarized by:
mean()
=

and
SE()
=

The shape will be approximately normal for sufficiently large samples.
This is reasonable to assume if each sample individually
meets the rule of thumb for the approximation.

Suppose that the proportion of hyperactive children in two separate populations
of retarded children is .40 for each.
Random samples of size 100 and 120 are respectively chosen.
What is the probability that the difference in the sample proportions,
()
is more than 16%?
The sampling distribution
for the difference in sample proportions
will be approximately normal with
a mean of 0
and a standard deviation of
= .0663.
The z-score is

z = (.16 - 0) / .0663 = 2.41

Since the area to the right of 2.41 under the standard normal curve
is .0080,
the probability is less than 1% that the difference would be this large.

Last modified: Feb 19, 1996

Bret Larget,
larget@mathcs.duq.edu