Consider the normal linear model where the $n$-vector of responses $Y$ satisfies $Y=X \beta+\varepsilon$ with $\varepsilon \sim N_{n}\left(0, \sigma^{2} I\right)$ and $X$ is an $n \times p$ design matrix with full column rank. Write down a $(1-\alpha)$-level confidence set for $\beta$.

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

In the model above with $n=100$ and $p=4$, you observe that one observation has Cook's distance 3.1. Would you be concerned about the influence of this observation? Justify your answer.

[Hint: You may find some of the following facts useful:

1. If $Z \sim \chi_{4}^{2}$, then $\mathbb{P}(Z \leqslant 1.06)=0.1, \mathbb{P}(Z \leqslant 7.78)=0.9$.

2. If $Z \sim F_{4,96}$, then $\mathbb{P}(Z \leqslant 0.26)=0.1, \mathbb{P}(Z \leqslant 2.00)=0.9$.

3. If $Z \sim F_{96,4}$, then $\left.\mathbb{P}(Z \leqslant 0.50)=0.1, \mathbb{P}(Z \leqslant 3.78)=0.9 .\right]$