I'm taking a regression analysis course and we were studying simple linear regression. I've understood how slope
$$ \hat\beta_1 follows \space N(0,\sigma^2 / S_{xx}) $$
and is normally disributed.
And
$$ \hat\sigma^2 follows \space \sigma^2 * \chi(n-2) / (n-2) $$
the estimated sigma^2 follows a chi square distribution.
but next he said that using this we can prove this term:
$$ (\hat\beta_1 - \beta_1)/se(\hat\beta_1) $$
Follows a t-distribution with n-2 degrees of freedom and I'm not able understand how do I go from a normal distribution to t-distribution using the above two equations.
Edit : So upon googling a bit there seems to be some relationship between chi square and t-distribution but I'm still not able to follow the logic behind that and how this one will work out.
