Suppose \(X_{1}, \ldots, X_{n}\) is a random sample from a \(\operatorname{Uniform}(0, \theta)\) density where \(\theta \in \Omega=(0, \infty)\).

a. Find a \(100 \alpha \%\) confidence interval for \(\theta\) when \(\omega_{U}-\omega_{L}=\alpha\) where \(\omega_{L} \in\) \([0,1]\) and \(\omega_{U} \in[0,1]\).

b. Use the confidence interval given above to derive an observed confidence level for an arbitrary region \(\Psi=\left(t_{L}, t_{U}ight) \subset \mathbb{R}\) where \(0