2. [8 marks] Consider the statistical model $$ Y_{i j}=\mu+\tau_{i}+R_{i j}, \quad R_{i j} \stackrel{\text { i.i.d }}{\sim} GO, \sigma) $$ where $i=1, \ldots, t, j=1, \ldots, r$, and $\sum_{i=1}^{t} \tau_{i}=0 $ Suppose $\theta=\sum_{i} a_{i} \tau_{i}$ is a contrast with the corresponding estimator $\widetilde{\theta}=\sum_{i} a_{i} \bar{Y}_{1+}$. Show that $$ \frac{\tilde{\theta}-\theta}{\widetilde{\sigma} \sqrt{\sum_{i} a_{i}^{2} / r}} $$ is a pivotal quantity and derive, with detailed steps, its distribution SP.PB. 109