In simple linear regression for the problem Y=aX+b; the "goodness" of the coefficient a can be estimated through a hypothesis test which computes the statistic, $t=\frac{a'-0}{SE(a')}$.
Which of the following follows a t distribution:
1- Is it t?
2- Is it a?
3- Is it both?
If one computed some other measure such as $t_2=\frac{2a'-0}{8SE(a')}$, would it also follow a t-distribution as long as it is computed on a sample? why/why not?