Simulation studies

The 2020 interval for a mean walking time, $\bar x \pm t_{n-1}\,s/\sqrt n$, is exact when the data are normal, because $(\bar X-\mu)/(S/\sqrt n)$ then has a t distribution. Real times are often skewed. This study draws samples from a normal, an exponential and a lognormal distribution and checks how often the interval catches the true mean.

The 2020 code estimated λ = σ² of log X by its MLE, $\hat\lambda = SS/n$, and then used the pivot built for the unbiased $S^2 = SS/(n-1)$:

The result is the textbook interval shrunk by (n − 1)/n. Its exact behaviour follows from $SS/\lambda\sim\chi^2_{n-1}$, and the simulation checks it on samples drawn under the lognormal model.

The council survey compared 520 of 800 with 600 of 1,000 using the Wald interval $\hat p_1-\hat p_2\pm z\,\widehat{\mathrm{se}}$. The Agresti–Caffo interval adds one success and one failure to each group before using the same formula. Exact coverage comes from summing binomial probabilities over every possible pair of counts, and the simulation runs both intervals on the same surveys.

The coursework compared two groups twice, once with a pooled t interval and once with a Wilcoxon rank-sum test on two weeks of 7 days. This study runs both tests at α = 5% on the same datasets, for normal, heavy-tailed and skewed data, shifting the second group by 0 to 1.5 standard deviations.

With 7 per group the exact Wilcoxon test cannot reject at exactly 5%. Its largest rejection region below 5% has probability 3.79%, its attainable size.

For an exponential sample of n = 50, the 2020 answer tested λ = 1 with an equal-tailed version of the likelihood-ratio test: reject when $Y=\sum X_i$ falls outside the 2.5% and 97.5% points of Gamma(50, 1). Its power at any λ then follows from Gamma(50, λ). Simulating the test directly is an independent check of that algebra.