Confidence intervals for a difference in population proportions

What the exam asks

Across multiple-choice and free-response questions, you must be ready to execute five specific tasks (Skills 2.C, 4.E, 3.E, 4.F, and 4.G):

1. Identify the procedure by name and define the parameter $p_1 - p_2$ in context, specifying the order of subtraction.
2. Verify the randomization, 10%, and normality conditions using actual numbers from the prompt.
3. Calculate the point estimate, standard error, margin of error, and confidence interval.
4. Interpret both the confidence interval and the confidence level in context.
5. Justify a claim about the two populations by checking whether $0$ (or another claimed difference) falls inside the interval.

Core ideas

Procedure and parameters

When estimating the difference between the proportions of successes in two distinct populations or treatment groups, the exact procedure name is a two-sample $z$-interval for a difference between population proportions.

Define both parameters clearly as population values (or, in an experiment, as the true values for the treatments) and state the direction of subtraction: let $p_1 - p_2$ represent the true difference in the proportion of [population 1] and [population 2] that have [characteristic of interest].

Conditions for inference

Before constructing the interval, check three conditions:

Formula, standard error, and margin of error

The confidence interval takes the general form $\text{point estimate} \pm \text{margin of error}$:

• Point estimate: $\hat{p}_1 - \hat{p}_2$