3.13 Carrying Out a Test for the Difference Between Two Population Proportions

Calculate an appropriate test statistic and p-value for testing a hypothesis for the difference between two population proportions.

• The test statistic for the difference between two population proportions is z pp pp nn cc   12 12 0 1 11 , where p np np nn c   1 1 2 2 12 is the proportion of successes for the two groups combined. The z -statistic has a standard normal distribution when the null hypothesis is true.
• The p-value for a two-sample z-test for the difference between two population proportions can be found from the standard normal distribution using a table or technology.

Interpret the p-value of a hypothesis test for the difference between two population proportions.

• The p-value is the probability of obtaining a test statistic as extreme or more extreme than the test statistic that was observed (i.e., in the direction of the alternative hypothesis) given that the null hypothesis is true. An interpretation of the p-value of a hypothesis test for a difference between two population proportions should include a statement that the p-value is computed assuming the null hypothesis is true (i.e., by assuming that the true population proportions are equal to each other in context).

Justify a claim about the populations based on the results of a hypothesis test for the difference between two population proportions.

• A formal decision in a hypothesis test for the difference between two population proportions explicitly compares the p-value to the significance level, α. If the p-value , then reject the null hypothesis, H: 0 pp12 or H: 01 2pp 0. If the p-value , then fail to reject the null hypothesis.
• The results of a hypothesis test for the difference between two population proportions can serve as the statistical reasoning to support the answer to an investigative question about the two populations that were sampled.
• A conclusion for the hypothesis test for the difference between two population proportions is stated in context consistent with, and in terms of, the alternative hypothesis using nondefinitive language. The conclusion should contain a reference to the parameters and the populations. Inference for Categorical Data: Proportions UNIT 3 110