Consider the linear regression model

$$Y_{i}=\beta x_{i}+\epsilon_{i},$$

$i=1, \ldots, n$, where $x_{1}, \ldots, x_{n}$ are given constants, and $\epsilon_{1}, \ldots, \epsilon_{n}$ are independent, identically distributed $N\left(0, \sigma^{2}\right)$, with $\sigma^{2}$ unknown.

Find the least squares estimator $\widehat{\beta}$ of $\beta$. State, without proof, the distribution of $\widehat{\beta}$ and describe how you would test $H_{0}: \beta=\beta_{0}$ against $H_{1}: \beta \neq \beta_{0}$, where $\beta_{0}$ is given.