2. Let $X$ be a random variable with distribution $$ P_{\theta}(X \leq x)=1-e^{-(x / \theta)^{2}} \quad \text { for } x \geq 0, $$ where $\theta>0$. (a) Consider testing $H_{0}: \theta \leq \theta_{0}$ versus $H_{1}: \theta>\theta_{0}$ for some positive constant $\theta_{0}$. We reject $H_{0}$ if $X>k$. What choice of $K$ would result in a size $\alpha$ test? (b) Consider the class of tests given in (a). Suppose that $\theta_{0}=2$ and the observed value of $X$ is $0.971$. What is the observed $p$-value? (c) Construct an $80 %$ confidence interval for $\theta$ that takes the form of $(0.2 X / C, X / c$ for some positive constant $c$. What value can $c$ take? You can use the fact that $(0.8043)(1-$ $\left. 0.8043^{25} ight)=0.8$. SP.JG. 148