I can take samples of a random variable $X \sim U(a, b)$, where the length of $(a, b)$ is known. I am interested in its mean $E[X]$, estimated with $\hat{X_n} = \frac{1}{n} \sum_{i=1}X_n$, but I need some guarantees for this estimate. Specifically, given a $\delta > 0$, I want to find out how many samples I should take s.t. $(\hat{X_n} - \delta, \hat{X_n} + \delta)$ is a 95% confidence interval.
I would know how to do this for a normally distributed random variable, but I'm not sure about the uniform case. Maybe a very similar or even the same procedure can be justified using the CLT?