11.1. Estimation and Hypothesis Testing for the Difference

between the Proportions of Two Populations

General Case:

: the proportion for population 1, : the proportion for population 2;

: the sample proportion for population 1,

: the sample proportion for population 2;

: the sample size for population 1, : the sample size for population 2.

The point estimate of : .

Important Properties of :

the sample statistic with possible values ,

the sample statistic with possible values

Then,

and

,

: the estimate of

Sampling Distribution of :

Suppose all greater or equal to 5.

Then,

confidence interval for :

Suppose all greater or equal to 5.

Then,

is a confidence interval.

Hypothesis Testing:

As , we can use the “pooled estimate”

,

to estimate and

to estimate .

Thus,

.

(I): vs.

Then,

In addition,

(II): vs.

Then,

In addition,

(III):

vs

Then,

In addition,

Example 1:

The results of a recent poll on the preference of voters regarding two candidates

are shown below:

Candidate	Voters Surveyed	Voters Favoring This Candidate
A	400	192
B	450	225

(a) Please construct a 90% confidence interval for the difference between the preference

for the two candidates ().

(b) With , please test whether or not there is a significant difference between

the preference for the two candidates ().

[solution:]

(a)

A 90% confidence interval for

(b)

Thus,

Since

,

we do not reject .

JavaStatSoft:

Z interval:

Statistics -> Estimation -> Two Samples -> Proportion and Count

Z test:

Statistics -> Tests -> Two Samples -> Proportion and Count