Assume that observations $Y=\left(Y_{1}, \ldots, Y_{n}\right)^{T}$ satisfy the linear model

$$Y=X \beta+\epsilon$$

where $X$ is an $n \times p$ matrix of known constants of full $\operatorname{rank} p<n$, where $\beta=\left(\beta_{1}, \ldots, \beta_{p}\right)^{T}$ is unknown and $\epsilon \sim N_{n}\left(0, \sigma^{2} I\right)$. Write down a $(1-\alpha)$-level confidence set for $\beta$.

Define Cook's distance for the observation $\left(x_{i}, Y_{i}\right)$, where $x_{i}^{T}$ is the $i$ th row of $X$. Give its interpretation in terms of confidence sets for $\beta$.

In the above model with $n=50$ and $p=2$, you observe that one observation has Cook's distance 1.3. Would you be concerned about the influence of this observation?

[You may find some of the following facts useful:

(i) If $Z \sim \chi_{2}^{2}$, then $\mathbb{P}(Z \leqslant 0.21)=0.1, \mathbb{P}(Z \leqslant 1.39)=0.5$ and $\mathbb{P}(Z \leqslant 4.61)=0.9$.

(ii) If $Z \sim F_{2,48}$, then $\mathbb{P}(Z \leqslant 0.11)=0.1, \mathbb{P}(Z \leqslant 0.70)=0.5$ and $\mathbb{P}(Z \leqslant 2.42)=0.9$.

(iii) If $Z \sim F_{48,2}$, then $\mathbb{P}(Z \leqslant 0.41)=0.1, \mathbb{P}(Z \leqslant 1.42)=0.5$ and $\mathbb{P}(Z \leqslant 9.47)=0.9$. ]