Compute p-values, CIs, and adjust multiples

Hypothesis testing and intervals in practice. Part A (z vs t): You sample n = 15 observations from a population with unknown variance and observe sample mean = 10.4 and sample standard deviation = 2.1. Test H0: mu = 9.5 vs two-sided alternative at alpha = 0.05. 1) State whether you must use a t-test or z-test and why. 2) Compute the test statistic and p-value. 3) Construct a 95% confidence interval for mu and interpret it. Part B (multiple testing): You perform 20 independent hypothesis tests at alpha = 0.05. 1) What is the expected number of false positives and the family-wise error rate if you do not adjust? 2) Apply Bonferroni to control FWER at 0.05; report the new per-test alpha. 3) In a one-way ANOVA with 4 groups where you wish to test all pairwise mean differences post hoc, explain when Tukey’s HSD is preferred over Bonferroni and what error rate it controls. Part C (errors and asymptotics): 1) Define Type I and Type II errors with a concrete example. 2) Explain CLT vs Law of Large Numbers, their assumptions, and give a scenario with heavy tails where CLT convergence is slow and how that affects inference.