Suppose $X_{1}, \ldots, X_{n}$ is a random sample from a normal population with mean $\mu_{1}$ and standard deviation $\sigma$. Another random sample $Y_{1}, \ldots, Y_{n}$ is selected from a normal population with mean $\mu_{2}$ and standard deviation $\sigma$. The two samples are independent. Find the distribution of $\bar{X}-\bar{Y}$ and use it to form a ratio that follows the $t$ distribution with $2 n-2$ degrees of freedom. Please show the entire derivation SP.AS366