(This is almost certainly covered in Statistics 101, but I missed that class..)

I have a real-world sampled signal $S[t]$ that is a constant $\hat{S}$ plus some noise $\epsilon[t]$. My goal is to find $\hat{S}$ with some high degree of confidence.

Intuitively: If I take one sample of $S$, I cannot extract $\hat{S}$. If I take an infinite amount of samples, I can perfectly reconstruct $\hat{S}$ (but that takes a while to compute ;). After $n$ samples, I can estimate $\hat{S}$, and my confidence in the estimation will increase as I take more samples.

I'd like to take enough samples so that the estimation of $\hat{S}$ is "good enough".

So: is there a function that describes the confidence in the estimate of $\hat{S}$ after $n$ samples?

addendum

From the comments, I realize I should have stated this up front:

The noise has a flat PDF. In other words, the noise is evenly distributed with some finite bounds. (It's clear why that makes a difference...)