An F-test involves the computation of an F-statistic, which is then compared to the critical values of an F-distribution for a given significance and numerator and denominator degrees-of-freedom.
An F-statistic is calculated by dividing a chi-squared distribution divided by its degrees-of-freedom by another (independent) chi-squared distribution by its degrees-of-freedom. The resulting F-statistic has two degrees-of-freedom parameters, one each for the numerator and the denominator.
Therefore, the F-statistic for would be:
We know (somehow) that , therefore we set the numerator equal to:
From the same implication of the last assumption of the CLRM as used by the t-test explanation,
Therefore, putting it all together gives us: