Confidence intervals for a difference in population means

What the exam asks

• Section I (Multiple Choice): Recognize the correct formula structure for the two-sample $t$-interval, verify conditions for both samples, interpret a given confidence interval or confidence level in context, or evaluate a claim based on whether the interval contains $0$ (or another specific difference).
• Section II (Free Response): Question 3 (or parts of Questions 2 and 4) may ask you to construct and interpret a complete confidence interval, earning points (each earned independently unless the scoring guideline says otherwise) for naming the procedure, defining the parameter, checking conditions, calculating the interval, and interpreting the result.

Core ideas

Procedure and parameter

When comparing a quantitative variable across two independent groups (as opposed to matched pairs), use a two-sample $t$-interval for a difference between two population means.

Define the parameter $\mu_1 - \mu_2$ in context, explicitly stating the order of subtraction:

• Observational study: $\mu_1 - \mu_2$ is the true difference in population means (Group 1 minus Group 2).
• Randomized experiment: $\mu_1 - \mu_2$ is the true difference in mean responses if all experimental units were assigned to Treatment 1 versus Treatment 2.

Conditions for inference

You must verify all three conditions for both groups using numbers and facts from the prompt:

Formula, standard error, and degrees of freedom

The confidence interval is calculated as:

• Point estimate: $\bar{x}_1 - \bar{x}_2$