Statistical Inference: Estimation and Hypothesis Testing
Core concepts
Know point estimation properties: unbiasedness, consistency, efficiency, and sufficiency. Understand method of moments, maximum likelihood, and the Cramér–Rao lower bound. Be able to construct confidence intervals for means, proportions, and variances, and to interpret them correctly. For testing, know the null and alternative, Type I and Type II errors, power, p-values, one- and two-tailed tests, and the standard tests: z, t, chi-square, F, and likelihood ratio tests. Understand the Neyman–Pearson lemma and the relationship between confidence intervals and hypothesis tests.
Traps
- Saying a 95% confidence interval contains the parameter with probability 0.95; the correct statement concerns repeated sampling.
- Interpreting a p-value as the probability that the null hypothesis is true.
- Using a z-test when the population variance is unknown and the sample is small; use t.
- Confusing statistical significance with practical importance.
- Forgetting that power increases with sample size and effect size, and decreases with stricter α.
- Choosing a one-tailed test after seeing the data.
Worked example
A sample of n=16 gives mean 52 and sample standard deviation 8. Test H0: μ=50 against H1: μ≠50 at α=0.05. t = (52−50)/(8/4) = 1.0 with 15 df; the critical value is about 2.131, so we fail to reject. Rule of thumb: the standard error of the mean is s/√n, and the t-distribution approaches the normal as df grows beyond about 30.