Statistical Inference
What the exam tests
Inference uses sample data to estimate population parameters and test hypotheses. Expect standard errors, confidence intervals, z- and t-tests, and interpretation of significance.
Core rules
- Estimator properties: unbiased if E(θ̂) = θ; efficient if it has smaller variance; consistent if it approaches θ as n grows.
- Standard error of the mean: SE = σ/√n. If σ is unknown and n is small, use s/√n with t and df = n − 1.
- Confidence interval for μ with known σ: x̄ ± z(α/2) σ/√n. Common values: 90% → 1.645, 95% → 1.96, 99% → 2.576.
- Hypothesis test: state H0 and H1; choose α; compute the test statistic; compare with the critical value or use the p-value.
- One-sample mean test: z = (x̄ − μ0)/(σ/√n) for known σ or large n; t = (x̄ − μ0)/(s/√n) for small n and unknown σ.
- Type I error: rejecting a true H0, probability α. Type II error: failing to reject a false H0.
- A two-tailed test splits α between both tails; a one-tailed test puts all α in one tail.
Worked example
n = 25, x̄ = 52, σ = 10. Test H0: μ = 50 against H1: μ ≠ 50 at 5%. SE = 10/5 = 2; z = (52 − 50)/2 = 1.00. Critical values are ±1.96; since |1.00| < 1.96, fail to reject H0. The 95% CI is 52 ± 1.96 × 2 = (48.08, 55.92), which contains 50 and agrees with the test.
Traps
Saying H0 is accepted rather than not rejected; using σ/√n as a variance; using z when n is small and σ is unknown; confusing one- and two-tailed critical values; reading p = 0.03 as a 3% chance that H0 is true; forgetting df = n − 1 for t.