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 ![{\displaystyle [Z(0,1)]^{2}=\chi ^{2}[1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/300819d8b54ba0b807a46c37dde4d27c29b72c14)
, therefore we set the numerator equal to:
![{\displaystyle Z({\hat {\beta _{1}}})^{2}={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}(\sum X_{i}^{2})}{\sigma ^{2}}}\sim \chi ^{2}[1]={\frac {\chi ^{2}[1]}{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8642e521196f03473d26ee811779039da882cbfc)
From the same implication of the last assumption of the CLRM as used by the t-test explanation,
![{\displaystyle {\frac {\chi ^{2}[N-2]}{N-2}}\sim {\frac {\hat {\sigma ^{2}}}{\sigma ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a88c5991932206ff9b95fb2f704caf99382acbf2)
Therefore, putting it all together gives us:
![{\displaystyle F({\hat {\beta _{1}}})={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}(\sum X_{i}^{2})/\sigma ^{2}}{{\hat {\sigma ^{2}}}/\sigma ^{2}}}={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}}{{\hat {\sigma ^{2}}}/\sum X_{i}^{2}}}={\frac {({\hat {\beta _{1}}}-\beta _{1})^{2}}{{\hat {Var}}({\hat {\beta _{1}}})}}\sim F[1,N-2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff093d1bc548f54afad3fd2a79d0cf097f92dcb2)
