We consider a test statistic $Z = \frac{| \bar{X} - \bar{Y} |}{\sigma} \sqrt{\frac{n}{2}}$, where
$$\bar{X} = \frac{1}{n}\sum_{i=1}^nX_i$$
$$\bar{Y} = \frac{1}{n}\sum_{i=1}^nY_i$$
$$X_i \sim N(\mu_1 , \sigma^2)$$
$$Y_i \sim N(\mu_2 , \sigma^2)$$
We're testing a zero hypothesis that $\mu_1 = \mu_2$ against the alternative that $\mu_1 \neq \mu_2$. I need to find Z's distribution, and since it's supposed to be a z-test, I expect it to be normal.
So, the way I see it, by CLT Z will follow normal distribution N(0,1) for big enough $n$ under zero hypothesis - but I'm not sure that it's actually the answer, since I think it should work for all values of $n$. The cause of my confusion is the absolute value - if we add or substract normal variables, the distribution will stay normal, only changing parameters. However absolute value of a normal variable doesn't have to be normal, and in fact I'm pretty sure that usually isn't.
I see 3 options here:
1) There's something really basic here that I just don't understand. 2) The CLT argument is actually enough explanation and it's supposed to only work for big samples. 3) The assumption of normality of Z is just wrong.
I expect that similar questions happened multiple times already, but for reasons unknown I couldn't find any.
The central limit theorem does not bear upon this, because the random variables you're starting with are already normally distributed. A linear combination of independent normally distributed random variables, in which the coefficients are not random variables but constants, is normally distributed. (You didn't mention independence, but I surmise that you intended that, and should have mentioned it.)
The central limit theorem treats of independent identically distributed random variables with finite variance that are not assumed to be normally distributed.