Probstat/notes/t-distributions

This is part of probstat.

In many applications including computing confidence intervals and also hypothesis testing, we need to know the distribution of the sample mean. Let ${\bar {X}}$ be a sample mean of a normal population computed from a sample of size $n$. If the variance $\sigma ^{2}$ of the population is known, we have that

${\frac {{\bar {X}}-\mu }{\sigma /{\sqrt {n}}}}\sim Normal(0,1).$

However, in most cases, we do not know $\sigma ^{2}$ and we only have the sample variance

$S^{2}={\frac {\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}}{n-1}}$.

Therefore, we would like to use the sample standard deviation $S$ instead of the real standard deviation $\sigma$. We hope that

${\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}}$

will be close to normal. This is indeed true. Although ${\sqrt {n}}({\bar {X}}-\mu )/S$ is not normal, it is t-distributed with $n-1$ degree of freedom.