(i) Consider the linear model

$$Y_{i}=\alpha+\beta x_{i}+\varepsilon_{i}$$

where observations $Y_{i}, i=1, \ldots, n$, depend on known explanatory variables $x_{i}$, $i=1, \ldots, n$, and independent $N\left(0, \sigma^{2}\right)$ random variables $\varepsilon_{i}, i=1, \ldots, n$.

Derive the maximum-likelihood estimators of $\alpha, \beta$ and $\sigma^{2}$.

Stating clearly any results you require about the distribution of the maximum-likelihood estimators of $\alpha, \beta$ and $\sigma^{2}$, explain how to construct a test of the hypothesis that $\alpha=0$ against an unrestricted alternative.

(ii) A simple ballistic theory predicts that the range of a gun fired at angle of elevation $\theta$ should be given by the formula

$$Y=\frac{V^{2}}{g} \sin 2 \theta$$

where $V$ is the muzzle velocity, and $g$ is the gravitational acceleration. Shells are fired at 9 different elevations, and the ranges observed are as follows:

$\begin{array}{cccccccccc}\theta \text { (degrees) } & 5 & 15 & 25 & 35 & 45 & 55 & 65 & 75 & 85 \\ \sin 2 \theta & 0.1736 & 0.5 & 0.7660 & 0.9397 & 1 & 0.9397 & 0.7660 & 0.5 & 0.1736 \\ Y(\mathrm{~m}) & 4322 & 11898 & 17485 & 20664 & 21296 & 19491 & 15572 & 10027 & 3458\end{array}$

The model

$$Y_{i}=\alpha+\beta \sin 2 \theta_{i}+\varepsilon_{i}$$

is proposed. Using the theory of part (i) above, find expressions for the maximumlikelihood estimators of $\alpha$ and $\beta$.

The $t$-test of the null hypothesis that $\alpha=0$ against an unrestricted alternative does not reject the null hypothesis. Would you be willing to accept the model $(*)$ ? Briefly explain your answer.

[You may need the following summary statistics of the data. If $x_{i}=\sin 2 \theta_{i}$, then $\bar{x} \equiv n^{-1} \sum x_{i}=0.63986, \bar{Y}=13802, S_{x x} \equiv \sum\left(x_{i}-\bar{x}\right)^{2}=0.81517, S_{x y}=\sum Y_{i}\left(x_{i}-\bar{x}\right)=$ 17186. ]