OLS - Why coefficient Beta has Normal Distribution but not t-Distribution I have a little trouble understanding the solution to the following problem. To my understanding, coefficients in OLS should follow t-distribution. However, the solution says it follows Normal.
Please see the answer in the picture


$\hat\beta_3 \sim N(\beta_3, 0.022 \sigma^2)$ is the distribution of the estimate $\hat\beta_3$ conditional on the values of $\beta_3$ and $\sigma$.
In inference, you often do not know $\beta_3$ and $\sigma$ and compute a statistic $\frac{\hat\beta_3}{\hat{\sigma}}$. That is the statistic which is t-distributed (if the null hypothesis, $\beta_3 = 0$, is true, otherwise it is non-central t-distributed).
For your 'problem 1' the idea is to ignore this for a while and just experience/understand/observe how the estimate $\hat\beta_3$ is distributed, when we know the actual values of $\beta_3$ and $\sigma$. It is a sort of thought experiment.
