T-test on a mean with unknown variance

Does a sample come from a population with mean $\mu_0$ ?

$$H_0:\mu = \mu_0$$ (6.3)

$$H_1: \mu \neq \mu_0 \nonumber$$

Since the population variance is no longer known we must estimate it using the sample variance $s^2$. This increases the uncertainty and modifies the sampling distribution of the test statistic slightly for small sample sizes $n<30$. Instead of being distributed normally, the test statistic is distributed as Student's t distribution $T\sim t_\nu$ with $\nu=n-1$ degrees of freedom. Student's t distribution has a density $f(t)\propto(1+t^2)^{-n/2}$, which resembles the normal density except that it has slightly fatter tails (leptokurtic) and so provides more chance of having values far from the mean. This test is often referred to as a one-sample t-test on the mean. Test using a T-score test statistic with the sampling distribution

$$T = \frac{\overline{X}-\mu_0}{s/\sqrt{n}}\sim t_{n-1}$$