Choose and Explain a Classical Hypothesis Test

Define the null and alternative hypotheses, significance level, p-value, Type I error, and Type II error. Then contrast the situations where a z-test, t-test, chi-square test, and F-test are suitable. For each one, identify the parameter or relationship under test, list its assumptions, and state what a statistically significant outcome does and does not prove.

Constraints & Assumptions

• Answer using frequentist definitions throughout.
• Keep the test statistic's reference distribution separate from the distribution of the observed data.
• Consider independence and distributional assumptions instead of choosing a test based only on sample size.
• Do not treat a p-value as the probability that the null hypothesis is true.

Clarifying Questions to Ask

• Is the outcome continuous, categorical, or an estimate of variance?
• Are the samples paired, independent, or grouped?
• Are normality and equal-variance assumptions reasonable, and is the variance known?

Hint: Start from the estimand. Decide on the test only after stating whether the target is a mean, proportion, independence relationship, or variance ratio.

Hint: State the reference world. A p-value describes how extreme the statistic is under the null model and its assumptions.

What a Strong Answer Covers

• Precise definitions and the link between alpha and Type I error under the null.
• Power and Type II error expressed as functions of effect size, variability, sample size, and test design.
• Correct use cases and assumptions for all four named test families.
• An interpretation that distinguishes statistical significance, effect size, and practical importance.

Follow-up Questions

• How would multiple testing alter the decision threshold?
• What would you report alongside a p-value to convey magnitude and uncertainty?

Overview: Define the null and alternative hypotheses, significance level, p-value, Type I error, and Type II error. Lay out the assumptions and derivation clearly, examine edge cases, and show how the conclusion shifts when those assumptions fail.