I have a sample of 100 items, each associated to a random variable for which I can compute expected value and variance: $X_1, X_2, ..., X_{100}$. From these, we can define the mean $\overline{X}=\frac{1}{100}\sum{X_i}$. I'd like to test the hypothesis $H_0:\mu=0$, (where $\mu$ is the true population mean from which the 100 items were sampled) but for that I need the variance of the sample of $X_i$'s.
So the question is: how do I compute the variance of the sample, given the individual expectations and variances of the random variables?
Thanks a lot!
EDIT: More info about the question
I have a set of 100 items, and there is a function that assigns a score to each of them. The problem is that computing that function is actually very expensive, so instead I have a process with which I can estimate the score: the more effort (e.g. money) I put into the process, the better the estimate (less variance). So initially the variance of each estimate is maximum, it decreases as I put more effort into the process, and eventually variance is 0 and expectation equals the actual score if I run the process completely (i.e. the original function).
Those 100 items represent just a random sample of a wider population of items, so I'd like to test the hypothesis of the population score being different from (or larger than) zero.