Statistics: If $X_1$ and $X_2$ are both normally distributed then explain why $X_1 - X_2$ can be standardized with mean 0 and standard deviation of 1 I am currently studying hypothesis testing for two populations and I would like a math major or someone experienced to explain to me why this particular statistic has a mean of 0 and a standard deviation of 1:
$$ z_{\bar{X_1}-\bar{X_2}} = \frac{\bar{X_1}-\bar{X_2} - \left(\mu_1 - \mu_2\right)}{\sigma_{\bar{X_1}-\bar{X_2}}}$$
The course that I'm taking is under the political science department. I see a lot of theoretical questions here and I would like it if someone can explain the fundamentals. How do you know that if $X_1$ and $X_2$ are both normally distributed then $X_1 - X_2$ is also normal? Why is it that when you standardize $X_1 - X_2$ you get the same mean and standard deviation like when $X_1$ or $X_2$ is standardized?

For every random variable $X$ with mean $\mu_X$ and standard deviation $\sigma_X$, the random variable $Z=(X-\mu_X)/\sigma_X$ has mean $0$ and standard deviation $1$. Hence all your questions are explained if, forgetting the normal distribution and nearly everything else, you can show that the mean of $\bar X_1-\bar X_2$ is $\mu_1-\mu_2$. Can you?
