Conditions for using a t-test for means: contradictory or not?

Question

Besides the conditions of independence and 10%, my teacher says that we need an approximately normal distribution before performing a t-test. My question is: Is this contradictory because the math we use in performing a t-test relies on a t-distribution which is not a normal distribution?

Answer 1

Not really. Though these conditions can be relaxed using central limit theorem-like arguments, the main situation for which the $t$ distribution is appropriate is when you are sampling from a normal distribution and you know neither its mean nor its variance. Then your test statistic when the null hypothesis is $E[X]=\mu$ is $\frac{\overline{X}-\mu}{S/\sqrt{n}}$ where $S$ is the sample standard deviation. This quantity is not normally distributed basically because occasionally the sample standard deviation is significantly smaller than the population standard deviation, and as a result this distribution has fatter tails than a $N(0,1)$ distribution.

A true normal test is basically useless because there is no real situation where you should exactly know the population standard deviation.