A t-test involves the computation of a t-statistic, which is then compared to the critical values of a t-distribution for a given significance level.
A t-test is essentially the Z-statistic of a variable divided by the square root of an independent chi-square distribution divided by its own degrees-of-freedom. The resulting value is the t-statistic with the same degrees-of-freedom as the chi-squared distribution.
Therefore, the t-statistic of would be:
We know (as an implication of the last assumption of the CLRM) that
Therefore, putting it all together we get,