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.