Combining the independent standard normal variable $\sqrt n(\overline X-\mu)/\sigma$ with the independent chi-squared variable $S_{XX}/\sigma^2$ gives
$$
\boxed{
\frac{\sqrt n(\overline X-\mu)}{\sqrt{S_{XX}/(n-1)}}
\sim t_{n-1}},
$$
the <Student's t-distribution> with $n-1$ degrees of freedom.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 20H / b / Solution

Let $t_{\nu,p}$ denote the $p$-quantile of <Student's t-distribution>. Part (b) gives the <Student t confidence interval>
$$
\boxed{\overline X\pm t_{n-1,1-\alpha/2}
\sqrt{\frac{S_{XX}}{n(n-1)}}}.
$$

Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 20H / c / ii / Solution

{c}
{wiki}

Student's t-test divides a normally distributed estimated effect by its estimated standard error. Under its null hypothesis the statistic has a <Student's t-distribution>, with degrees of freedom determined by the variance estimate.

Student's t-test

{c}
{wiki}

Student's t-distribution with $\nu$ degrees of freedom is the distribution of $Z/\sqrt{V/\nu}$ for independent $Z\sim N(0,1)$ and $V\sim\chi^2_\nu$.


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