From each of 3 populations, $n$ data points are sampled and these are believed to obey

$$y_{i j}=\alpha_{i}+\beta_{i}\left(x_{i j}-\bar{x}_{i}\right)+\epsilon_{i j}, \quad j \in\{1, \ldots, n\}, i \in\{1,2,3\},$$

where $\bar{x}_{i}=(1 / n) \sum_{j} x_{i j}$, the $\epsilon_{i j}$ are independent and identically distributed as $N\left(0, \sigma^{2}\right)$, and $\sigma^{2}$ is unknown. Let $\bar{y}_{i}=(1 / n) \sum_{j} y_{i j}$.

(i) Find expressions for $\hat{\alpha}_{i}$ and $\hat{\beta}_{i}$, the least squares estimates of $\alpha_{i}$ and $\beta_{i}$.

(ii) What are the distributions of $\hat{\alpha}_{i}$ and $\hat{\beta}_{i}$ ?

(iii) Show that the residual sum of squares, $R_{1}$, is given by

$$R_{1}=\sum_{i=1}^{3}\left[\sum_{j=1}^{n}\left(y_{i j}-\bar{y}_{i}\right)^{2}-\hat{\beta}_{i}^{2} \sum_{j=1}^{n}\left(x_{i j}-\bar{x}_{i}\right)^{2}\right]$$

Calculate $R_{1}$ when $n=9,\left\{\hat{\alpha}_{i}\right\}_{i=1}^{3}=\{1.6,0.6,0.8\},\left\{\hat{\beta}_{i}\right\}_{i=1}^{3}=\{2,1,1\}$,

$$\left\{\sum_{j=1}^{9}\left(y_{i j}-\bar{y}_{i}\right)^{2}\right\}_{i=1}^{3}=\{138,82,63\}, \quad\left\{\sum_{j=1}^{9}\left(x_{i j}-\bar{x}_{i}\right)^{2}\right\}_{i=1}^{3}=\{30,60,40\}$$

(iv) $H_{0}$ is the hypothesis that $\alpha_{1}=\alpha_{2}=\alpha_{3}$. Find an expression for the maximum likelihood estimator of $\alpha_{1}$ under the assumption that $H_{0}$ is true. Calculate its value for the above data.

(v) Explain (stating without proof any relevant theory) the rationale for a statistic which can be referred to an $F$ distribution to test $H_{0}$ against the alternative that it is not true. What should be the degrees of freedom of this $F$ distribution? What would be the outcome of a size $0.05$ test of $H_{0}$ with the above data?