3.12 Setting Up a Test for the Difference Between Two Population Proportions

Identify an appropriate testing method for the difference between population proportions including the parameters.

• The appropriate testing method for the difference between two population proportions is a two-sample z-test for the difference between two population proportions.
• The parameters for a hypothesis test for the difference between two population proportions should reference the population parameters, the response variables, and the populations in context.

Identify the null and alternative hypotheses for the difference between population proportions.

• For a two-sample z-test for the difference between two population proportions, the null hypothesis indicates no difference. The null hypothesis for the difference between two population proportions can be written as either H: 01 2pp or H: 001 2pp . A onesided alternative hypothesis for the difference bet ween two population proportions can be written as either H: a pp12 or equivalently H: a pp12 0, or H: a pp12 or equivalently H: a pp12 0. A two-sided alternative hypothesis for the difference bet ween two population proportions can be written as either H: a pp12 or equivalently H: a pp12 0 .

Justify the appropriateness of a hypothesis test for the difference between two population proportions by verifying conditions.

• A two-sample z-test for a difference between two population proportions requires that three conditions be met:
  • i. The randomization condition—the data should be collected using two independent random samples or a randomized experiment.
  • ii. The 10% condition—when sampling without replacement, the size of each sample should be less than or equal to 10% of the respective population size: n s t N11 10% and n2 ulati N210% , where N1 i he size of pop on 1 and N2 is the size of population 2. The sample sizes are represented as n1 and n2. (Note: This condition is unnecessary when the data are from a randomized experiment.)
  • iii. The normality condition—n pc1 , n pc1 1  , n pc2  , and n pc2 1  , must all be at least 10, with p np np nn c   1 1 2 2 12 being the combined (or pooled) proportion assuming that H0 is true (H01: pp 2 or H:01 pp 2 0). Inference for Categorical Data: Proportions UNIT 3 108