Chapter 26

One generates a number \(x\) from a uniform distribution on the interval \([0, \theta]\). One decides to test \(H_{0}: \theta=2\) against \(H_{1}: \theta \neq 2\) by rejecting \(H_{0}\) if \(x \leq 0.1\) or \(x \geq 1.9 .\) a. Compute the probability of committing a type I error. b. Compute the probability of committing a type II error if the true value of \(\theta\) is \(2.5\).

To investigate the hypothesis that a horse's chances of winning an eighthorse race on a circular track are affected by its position in the starting lineup the starting position of each of 144 winners was recorded ([30]). It turned out that 29 of these winners had starting position one (closest to the rail on the inside track). We model the number of winners with starting position one by a random variable \(T\) with a \(\operatorname{Bin}(144, p)\) distribution. We test the hypothesis \(H_{0}: p=1 / 8\) against \(H_{1}: p>1 / 8\) at level \(\alpha=0.01\) with \(T\) as test statistic. a. Argue whether the test procedure involves a right critical value, a left critical value, or both. b. Use the normal approximation to compute the critical value(s) corresponding to \(\alpha=0.01\), determine the critical region, and report your conclusion about the null hypothesis.

This exercise is meant to illustrate that the shape of the critical region is not necessarily similar to the type of alternative hypothesis. The type of alternative hypothesis and the test statistic used determine the shape of the critical region. Suppose that \(X_{1}, X_{2}, \ldots, X_{n}\) form a random sample from an \(\operatorname{Exp}(\lambda)\) distribution, and we test \(H_{0}: \lambda=1\) with test statistics \(T=\bar{X}_{n}\) and \(T^{\prime}=\mathrm{e}^{-\bar{X}_{n}}\). a. Suppose we test the null hypothesis against \(H_{1}: \lambda>1\). Determine for both test procedures whether they involve a right critical value, a left critical value, or both. b. Same question as in part a, but now test against \(H_{1}: \lambda \neq 1\).