Given $N$ samples $x_1,\dots, x_n$ with mean $\bar x$ and variance $s^2$ drawn from a normally distributed process with mean $\mu$ and variance $\sigma^2$. I also have a variable $0 < p < 1,$
I would like to determine the value $v$ such that $P(x_{n+1} > v) = p$
As $N$ goes to infinity, $v$ goes to the result of the corresponding quantile function for a normal random variable with mean $\bar x$ and variance $s^2,$ but if $N$ is small enough, the fact that $\bar x \ne\mu$ and $s^s\ne\sigma^2$ has to be taken into account. How exactly do I do that?