Suppose that $X$ is a single observation drawn from the uniform distribution on the interval $[\theta-10, \theta+10]$, where $\theta$ is unknown and might be any real number. Given $\theta_{0} \neq 20$ we wish to test $H_{0}: \theta=\theta_{0}$ against $H_{1}: \theta=20$. Let $\phi\left(\theta_{0}\right)$ be the test which accepts $H_{0}$ if and only if $X \in A\left(\theta_{0}\right)$, where

$$A\left(\theta_{0}\right)= \begin{cases}{\left[\theta_{0}-8, \infty\right),} & \theta_{0}>20 \\ \left(-\infty, \theta_{0}+8\right], & \theta_{0}<20\end{cases}$$

Show that this test has size $\alpha=0.10$.

Now consider

$$\begin{aligned} &C_{1}(X)=\{\theta: X \in A(\theta)\} \\ &C_{2}(X)=\{\theta: X-9 \leqslant \theta \leqslant X+9\} \end{aligned}$$

Prove that both $C_{1}(X)$ and $C_{2}(X)$ specify $90 \%$ confidence intervals for $\theta$. Find the confidence interval specified by $C_{1}(X)$ when $X=0$.

Let $L_{i}(X)$ be the length of the confidence interval specified by $C_{i}(X)$. Let $\beta\left(\theta_{0}\right)$ be the probability of the Type II error of $\phi\left(\theta_{0}\right)$. Show that

$$E\left[L_{1}(X) \mid \theta=20\right]=E\left[\int_{-\infty}^{\infty} 1_{\left\{\theta_{0} \in C_{1}(X)\right\}} d \theta_{0} \mid \theta=20\right]=\int_{-\infty}^{\infty} \beta\left(\theta_{0}\right) d \theta_{0}$$

Here $1_{\{B\}}$ is an indicator variable for event $B$. The expectation is over $X$. [Orders of integration and expectation can be interchanged.]

Use what you know about constructing best tests to explain which of the two confidence intervals has the smaller expected length when $\theta=20$.