Hypothesis Testing and Inference
What this domain tests
Forming H0 and H1, choosing the correct statistic, running one- and two-tailed tests at a stated significance level, interpreting p-values and error types, plus chi-squared goodness-of-fit and independence tests and inference about correlation.
Core rules
- H0 always states equality (μ = μ0); H1 fixes the direction (> , < or ≠), taken from the question, not the data.
- Mean test with known σ, or large n: z = (x̄ - μ0)/(σ/√n). Unknown σ with small n: t with n - 1 degrees of freedom.
- Reject H0 when the statistic lies in the critical region, equivalently when p < α.
- Type I error: rejecting a true H0 (probability α). Type II error: failing to reject a false H0.
- Chi-squared: χ² = Σ(O - E)²/E, with df = (r - 1)(c - 1) for independence and k - 1 - m for goodness of fit when m parameters are estimated. Combine expected frequencies below 5.
Worked example
A sample of 36 gives x̄ = 52. Test H0: μ = 48 against H1: μ > 48 at the 5% level, with σ = 12. z = (52 - 48)/(12/√36) = 4/2 = 2.00. For a one-tailed 5% test the critical value is 1.645; since 2.00 > 1.645, reject H0. The p-value is P(Z > 2.00) = 0.0228, less than 0.05, confirming the decision: there is sufficient evidence that the mean exceeds 48.
Traps
- Using 1.96 for a one-tailed 5% test.
- Writing 'accept H0' instead of 'do not reject H0', with no conclusion in context.
- Wrong degrees of freedom in chi-squared, especially after estimating a parameter.
- Deciding the direction of H1 after inspecting the sample mean.