Two series of experiments are performed, the first resulting in observations $X_{1}, \ldots, X_{m}$, the second resulting in observations $Y_{1}, \ldots, Y_{n}$. We assume that all observations are independent and normally distributed, with unknown means $\mu_{X}$ in the first series and $\mu_{Y}$ in the second series. We assume further that the variances of the observations are unknown but are all equal.

Write down the distributions of the sample mean $\bar{X}=m^{-1} \sum_{i=1}^{m} X_{i}$ and sum of squares $S_{X X}=\sum_{i=1}^{m}\left(X_{i}-\bar{X}\right)^{2}$.

Hence obtain a statistic $T(X, Y)$ to test the hypothesis $H_{0}: \mu_{X}=\mu_{Y}$ against $H_{1}: \mu_{X}>\mu_{Y}$ and derive its distribution under $H_{0}$. Explain how you would carry out a test of size $\alpha=1 / 100$.