3.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions

Identify an appropriate confidence interval procedure including the parameters for the difference between two population proportions.

• Based on the sample data, a confidence interval can be calculated to estimate the difference between two population proportions. The appropriate confidence interval procedure is a two-sample z-interval for a difference between population proportions.
• The parameters of a confidence interval for the difference between two population proportions should refer to the difference in the proportions, the response variable, and the populations in context.

Justify the appropriateness of constructing a confidence interval for the difference between two population proportions by verifying conditions.

• A two-sample z-interval 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 t N11 10≤ % and n ula N22 10≤ % , where N1 is he size of pop tion 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—the number of observed successes, n p1 1  and n p2 2  , and number of observed failures, n p1 11  and n p2 21  , for both samples are all at least 10.

Calculate an appropriate confidence interval for the difference between two population proportions.

• The point estimate for the difference between two population proportions is pp 12 .
• For the difference between two population proportions, the interval estimate can be constructed as point estimate (margin of error). The interval estimate for the difference between two population proportions is  ( )     112 12 2 12 1 1p pp p pp z nn .

Calculate the standard error and margin of error for estimating the difference between two population proportions. Inference for Categorical Data: Proportions UNIT 3 104

• The standard error (SE ) for the difference between two population proportions is SE pp n pp npp   12 1 1 2 2 1211 .
• For the difference between two population proportions, the margin of error is the critical value z times the standard error (SE) of the difference between the two proportions, which equals z pp n pp n   1 1 2 2 1211 .