(a) Define a $100 \gamma \%$ confidence interval for an unknown parameter $\theta$.

(b) Let $X_{1}, \ldots, X_{n}$ be i.i.d. random variables with distribution $N(\mu, 1)$ with $\mu$ unknown. Find a $95 \%$ confidence interval for $\mu$.

[You may use the fact that $\Phi(1.96) \simeq 0.975 .]$

(c) Let $U_{1}, U_{2}$ be independent $U[\theta-1, \theta+1]$ with $\theta$ to be estimated. Find a $50 \%$ confidence interval for $\theta$.

Suppose that we have two observations $u_{1}=10$ and $u_{2}=11.5$. What might be a better interval to report in this case?