Confidence intervals using the t-distribution

t-tests and confidence intervals

For a small sample from a normal population with unknown variance, $\frac{\bar X-\mu}{S/\sqrt n}$ follows a $t$-distribution with $n-1$ degrees of freedom. It has heavier tails than $N(0,1)$, so intervals are wider: $\bar x\pm t_{n-1}\frac s{\sqrt n}$.