Student t from two independent samples

Let $X_{1},X_{2},…,X_{n}$ be a random sample from $N(\mu_{1},\sigma^{2})$ distribution and $Y_{1},Y_{2},…,Y_{m}$ be a random sample from $N(\mu_{2},\sigma^{2})$ distribution, where $\mu_{i}\in\mathbb{R}$, $i=1,2$ and $\sigma>0$. Suppose that the two random samples are independent. Define $$\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}\quad\text{and}\quad W=\frac{\sqrt{mn}\,(\bar{X}-\mu_{1})}{\sqrt{\sum_{i=1}^{m}(Y_{i}-\mu_{2})^{2}}}$$ Then which one of the following statements is TRUE for all positive integers $m$ and $n$?