Tests for a difference in population means

What the exam asks

• Section I (Multiple choice): You may be asked to identify the correct test statistic expression from summary statistics, select the proper hypotheses and parameter definitions, check conditions for two independent groups, or interpret a given $p$-value in context.
• Section II (Free response, for example Question 3): You may be asked to execute a complete hypothesis test from start to finish—naming the procedure, stating hypotheses, defining parameters, verifying conditions for both groups, calculating $t$ and the $p$-value using technology, and stating a contextual conclusion.

Core ideas

1. Identifying the procedure and parameters

Use a two-sample $t$-test for a difference between two population means when comparing a quantitative response variable across two independent groups (either two independent random samples or two groups formed by random assignment in an experiment).

Always define both parameters $\mu_1$ and $\mu_2$ in context (or define $\mu_1 - \mu_2$ with the order of subtraction clearly specified). For experiments, define as the true mean response if all experimental units like those in the study were to receive that treatment.

2. Hypotheses

The null hypothesis states that there is no difference between the two population (or treatment) means:

• $H_0: \mu_1 - \mu_2 = 0$ (or )

The alternative hypothesis reflects the research question:

• One-sided: $H_a: \mu_1 - \mu_2 > 0$ (or ) or (or )

3. Conditions for inference

You must verify and state the conditions for both samples/groups using specific numbers from the prompt: