Let ${Y_1,...,Y_n}$ be independent random variables and $Y_i$~$N(\beta x_i, 1)$ where $x_1,...,x_n$ are fixed known constants, and $\beta$ is an unknown parameter.
I found the MLE $\hat{\beta}$ is $$\frac{\sum^n_{i=1}y_ix_i}{\sum^n_{i=1}x_i^2}$$
Why is $$\sqrt{\sum^n_{i=1}x_i^2}\left(\hat{\beta}-\beta\right) \sim N(0, 1)$$
All I need is the general procedure written in words to get started.