(Solved): Consider the single sample mean model, which has the form \[ y_{i}=\mu+e_{i}, \quad i=1,2, \ldots, ...

Consider the single sample mean model, which has the form \[ y_{i}=\mu+e_{i}, \quad i=1,2, \ldots, n, \] where the standard assumptions are that \( E\left(e_{i}\right)=0, \operatorname{var}\left(e_{i}\right)=\sigma^{2} \), and \( \operatorname{cov}\left(e_{i}, e_{j}\right)=0 \) if \( i \neq j \). The least squares estimate of \( \mu \) minimizes the residual sum of squares, \[ R S S\left(\beta_{1}\right)=\sum_{i=1}^{n}\left(y_{i}-\mu\right)^{2} \] as a function of \( \mu \). (a) (2 pts) Derive a formula for the least squares estimate \( \hat{\mu} \) for \( \mu \). (Hint: take the derivative of \( R S S(\mu) \) with respect to \( \mu \) and determine where the derivative is zero). (b) (2 pts) Show that \( \hat{\mu} \) is an unbiased estimator of \( \mu \) under the standard assumptions given. (c) (2 pts) Show that \[ \operatorname{var}(\hat{\mu})=\frac{\sigma^{2}}{n} \] (d) (2 pts) Under the assumptions given, find \( E\left(y_{1}\right) \) and \( \operatorname{var}\left(y_{1}\right) \) (e) (2 pts) For the model given above, \( \hat{y}_{i}=\hat{\mu} \), for \( i=1,2, \ldots, n \). Under the assumptions given, find expressions for \( E\left(\hat{y}_{1}\right) \) and \( \operatorname{var}\left(\hat{y}_{1}\right) \).