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.

${\displaystyle t={\frac {Z}{\sqrt {V/m}}}\sim t[m]}$

Therefore, the t-statistic of ${\displaystyle \beta _{1}}$ would be:

• Numerator:

${\displaystyle Z({\hat {\beta _{1}}})={\frac {{\hat {\beta _{1}}}-\beta _{1}}{se({\hat {\beta _{1}}})}}={\frac {({\hat {\beta _{1}}}-\beta _{1})(\sum X_{i}^{2})^{1}/2}{\sigma }}}$

• Denominator:

We know (as an implication of the last assumption of the CLRM) that ${\displaystyle {\frac {(N-2){\hat {\sigma ^{2}}}}{\sigma ^{2}}}\sim \chi ^{2}[N-2]}$

Therefore, ${\displaystyle {\frac {\hat {\sigma ^{2}}}{\sigma ^{2}}}\sim {\frac {\chi ^{2}[N-2]}{[N-2]}}\Rightarrow {\sqrt {\frac {\chi ^{2}[N-2}{[N-2]}}}\sim {\frac {\hat {\sigma }}{\sigma }}}$

Therefore, putting it all together we get,

${\displaystyle t({\hat {\beta _{1}}})={\frac {Z({\hat {\beta _{1}}})}{{\hat {\sigma }}/\sigma }}={\frac {({\hat {\beta _{1}-\beta _{1}}})(\sum X_{i}^{2})^{1/2}/\sigma }{\sigma ^{2}/\sigma }}={\frac {{\hat {\beta _{1}}}-\beta _{1}}{{\hat {\sigma }}/(\sum X_{i}^{2})^{1/2}}}={\frac {{\hat {\beta _{1}}}-\beta _{1}}{{\hat {se}}({\hat {\beta _{1}}})}}\sim t[N-2]}$

Notes

• ${\displaystyle se({\hat {\beta _{1}}})={\frac {\sigma }{(\sum X_{i}^{2})^{1/2}}}}$
• ${\displaystyle {\hat {se}}({\hat {\beta _{1}}})={\frac {\hat {\sigma }}{(\sum X_{i}^{2})^{1/2}}}}$