Suppose $X_1,...,X_n$ are iid from $N(\mu,\sigma_1^2)$, $Y_1,...,Y_m$ are iid from $N(\mu,\sigma_2^2)$. $X_i$ are independent of $Y_j$. Both $\sigma_1$ and $\sigma_2$ are unknown. How to calculate CI for $\mu$?
Because two variances are unknown, and $n$ and $m$ are not necessarily equal, It's hard to find an appropriate statistic.