I'd like to understand the following problem, which comes from a real situation. Suppose I have a population of $N$ elements, over which a random variable $X$ is defined. To clarify ideas, this $N$ is not large ($N = 59$ in the concrete situation).
Suppose now I have a sample of $n$ individuals of the population, over which $X$ has mean $\mu$ and standard deviation $\sigma$. (To fix ideas, $n = 15$ in the concrete situation).
What is the best estimate of the mean and standard deviation of the whole population given $\mu$ and $\sigma$?
In other words, what is the probability that the real mean is in some (specific) interval around $\mu$? Same question for the relation between real standard deviation and $\sigma$.
Standard formulas work for very large population in relation to the sample, in my case I would like something more precise, say, a formula for the standard deviation of the mean which should tend to zero if $n$ tends to $N$...