Compute p-values, CIs, and adjust multiples

Part A (z vs t): You draw a sample of n = 15 observations from a population whose variance is unknown, and you observe a sample mean of 10.4 with a sample standard deviation of 2.1. Test H0: mu = 9.5 against a two-sided alternative at alpha = 0.05. 1) Say whether a t-test or a z-test is required and justify your choice. 2) Calculate the test statistic and the p-value. 3) Build a 95% confidence interval for mu and explain what it means. Part B (multiple testing): You run 20 independent hypothesis tests at alpha = 0.05. 1) What is the expected number of false positives, and what is the family-wise error rate if no adjustment is made? 2) Use Bonferroni to keep FWER at 0.05; state the new per-test alpha. 3) In a one-way ANOVA with 4 groups where you plan to test every pairwise mean difference post hoc, explain when Tukey's HSD is preferable to Bonferroni and which error rate it controls. Part C (errors and asymptotics): 1) Define Type I and Type II errors and give a concrete example. 2) Explain the CLT versus the Law of Large Numbers, their assumptions, and describe a heavy-tailed scenario where CLT convergence is slow and how that impacts inference.

Overview: This question tests a candidate's command of statistical inference, spanning hypothesis testing and test selection, confidence interval construction, multiple-testing corrections, error types, and asymptotic reasoning.