Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / O / 3 / a / Solution

For a normal sample with unknown mean and variance, the sample mean and sample variance are independent, with $\sqrt{n}(\overline{X}-\mu )/\sigma \sim N(0,1)$ and $(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$. Therefore the ratio defining the Studentized normal sample mean hasusing $S^2=(n-1{)}^{-1}{\sum }_i(X_i-\overline{X}{)}^2$.