Inference For Proportions:

The proportion of a population having a given characteristic is a parameter, p.  The proportion of a sample having a given characteristic is a statistic, .

We know that the sampling distribution of  is approximately normal (for sufficiently large samples) with mean  and standard deviation .

To standardize , subtract the mean and divide by the standard deviation.  This gives the z test statistic:

For a hypothesis test where , use  to estimate .

For a confidence interval, use  as an estimate of .

Assumptions for inference about a proportion:

1.         SRS

2.         population ³ 10n

3.         ³ 10 and  ³ 10 for a hypothesis test

³ 10 and  ³ 10 for a confidence interval

The margin of error is:

So to determine the sample size needed for a specified margin of error, let

and solve for n.

If we want to compare two populations or compare the responses to two treatments from independent samples, we look at a two-sample proportion.

The null hypothesis is that there is no difference between the two parameters.

or

The alternative hypothesis could be that

(two-sided)

or      (one-sided)

or      (one-sided)

Before you begin, check your assumptions!

Assumptions for inference about a proportion:

1.         SRS

2.         independently chosen samples

3.         population ³ 10n

4.         , , ,  ³ 5 for a hypothesis test

, , ,  ³ 5 for a C.I.

If these assumptions hold, then the difference in sample proportions is an unbiased estimator of the difference in population proportions, so   is equal to .

Also, the variance of  is the sum of the variances of  and , which is .

In order to standardize , subtract the mean and divide by the standard error:

Use this test statistic to carry out a test of significance.

On the TI-83+, use 2-PropZInt and 2-PropZTest to construct confidence intervals and perform significance tests.