Statistical Inference (Estimation and Hypothesis Testing)
Statistical inference asks whether sample evidence supports a claim about a population. The exam expects a complete test: hypotheses, significance level, test statistic, critical region or p-value, decision, and a conclusion phrased in context. Confidence intervals for a population mean also appear.
Sampling and standard error
- The sample mean has E(X̄) = μ and Var(X̄) = σ²/n, so its standard error is σ/√n.
- For large n (roughly n ≥ 30) the central limit theorem makes X̄ approximately normal whatever the population shape.
- Confidence interval for μ with known σ: x̄ ± z σ/√n, where z = 1.96 for 95% and 2.576 for 99%.
Hypothesis testing
- State H₀ as an equality and H₁ as >, < or ≠, matching the wording of the question.
- Choose the statistic: z = (x̄ − μ)/(σ/√n) for a mean, or a binomial or Poisson count.
- Compare with the critical value at the stated significance level, or compare the p-value with α.
- Decide, then conclude in the context of the problem, for example "there is insufficient evidence at the 5% level that the mean exceeds 12".
Worked example
A sample of 50 items has mean 12.4, and the population standard deviation is 2. Test at the 5% level H₀: μ = 12 against H₁: μ > 12. z = (12.4 − 12)/(2/√50) = 0.4/0.2828 = 1.414. The one-tailed 5% critical value is 1.645. Since 1.414 < 1.645, H₀ is not rejected; there is insufficient evidence at the 5% level that the population mean exceeds 12.
Traps
- Using 1.96 for a one-tailed 5% test instead of 1.645.
- Dividing by σ rather than σ/√n when standardising X̄.
- Writing "H₀ is accepted" rather than "H₀ is not rejected", or omitting the contextual conclusion.
- Mixing up errors: Type I is rejecting a true H₀, Type II is failing to reject a false H₀.
- Letting a one-tailed alternative use a two-tailed critical region.