Statistical Inference: Estimation and Hypothesis Testing
Inference uses sample evidence to say something about a population. The exam tests properties of good estimators, confidence intervals, and the formal machinery of a test: hypotheses, test statistic, critical region, errors and conclusion.
Estimation
- A point estimate is a single value; an interval estimate gives a range with a confidence level.
- Desirable properties: unbiasedness (E(θ̂) = θ), consistency (θ̂ → θ as n grows), efficiency (smaller variance among unbiased estimators).
- Confidence interval for a mean with known σ: x̄ ± z(σ/√n), using z = 1.96 for 95% and 2.58 for 99%. For small samples with σ unknown, use t with n − 1 degrees of freedom.
Hypothesis testing
Steps: state H₀ and H₁; choose the significance level α; select the test statistic; find the critical value for the tail(s); compare and conclude. For a large-sample mean, z = (x̄ − μ₀)/(σ/√n).
- Type I error: rejecting H₀ when true, probability α. Type II error: not rejecting a false H₀, probability β; power = 1 − β.
- One-tailed tests place all α in one tail (z = 1.645 at 5%); two-tailed tests split it (z = 1.96).
Worked example
With n = 100, x̄ = 52, σ = 10, test H₀: μ = 50 against H₁: μ > 50 at 5%. z = (52 − 50)/(10/10) = 2.00, which exceeds 1.645, so H₀ is rejected. The 95% two-tailed interval, 52 ± 1.96(1) = (50.04, 53.96), likewise excludes 50.
Traps that cost marks
Saying H₀ is “accepted” rather than “not rejected”; using 1.96 for a one-tailed test; interchanging Type I and Type II errors; using σ/√n with s and small n instead of the t statistic; misreading a p-value as the probability that H₀ is true.