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

#### Key Concepts

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.

#### Example

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

```  H0: p1 = p2
HA: p1 does not equal p2
```
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 = (x1 + x2) / (n1 + n2)
```
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.