Robust Variance of the Sample Mean Under Sampling Without Replacement

An urn contains $N$ balls numbered $1, 2, \dots, N$. You draw $n$ balls without replacement and let $\bar{X} = \tfrac{1}{n}\sum_{i=1}^{n} X_i$, where $X_i$ is the number on the $i$-th draw. Derive $\operatorname{Var}(\bar{X})$ in terms of $N$ and $n$, and evaluate it for $N = 10$, $n = 4$.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.