If $X_{1}, \ldots, X_{n}$ is a sample from a density $f(\cdot \mid \theta)$ with $\theta$ unknown, what is a $95 \%$ confidence set for $\theta$ ?

In the case where the $X_{i}$ are independent $N\left(\mu, \sigma^{2}\right)$ random variables with $\sigma^{2}$ known, $\mu$ unknown, find (in terms of $\sigma^{2}$ ) how large the size $n$ of the sample must be in order for there to exist a $95 \%$ confidence interval for $\mu$ of length no more than some given $\varepsilon>0$.

[Hint: If $Z \sim N(0,1)$ then $P(Z>1.960)=0.025 .]$