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 10n

3. 10 and 10 for a hypothesis test

10 and 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 10n

4. , , , 5 for a hypothesis test

, , , 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.