I am trying to prove that
$T_n=\frac{\bar{X}_n - \mu}{S/\sqrt{n}}\sim t_{n-1}$.
One of the assumptions that seems to come up in proofs I saw of this is that
$\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1)$.
and I was wondering why this is the case. I thought this result is approximate, and comes from the central limit theorem, and thus that the sample size must be "large enough". Then why, if this is the case, do we use the t-statistic, which uses this fact, when the sample size is "small".
It's been a while, but I believe that CLT-result is exact when you're dealing with actual normal samples. (In other words, if you're drawing from say, a uniform distribution, yes it is an approximation to the distribution of the average, but when you draw normally, you get exactly that.) This is probably easily shown via moment-generating functions.