For the model $Y = α + βx + \epsilon$, estimate mean and predicted value of Y when $x = x_0$ Consider n independent observations {($x_i, y_i$) : $1 \le i \le n$} from the model $$Y = α + βx + \epsilon,$$
where $\epsilon$ is normal with mean 0 and variance $σ^2$. Let $\hat α$, $\hat β$ and $\hat σ^2$ be the maximum likelihood estimators of $α$, $β$ and $σ^2$ respectively. Let $v_{11}$,
$v_{22}$ and $v_{12}$ be the estimated values of Var($\hat α$), Var($\hat β$) and Cov($\hat α$,$\hat β$) respectively. Then
(a) What is the estimated mean of $Y$ when $x = x_0$? Estimate the
mean squared error of this estimator.
(b) What is the predicted value of $Y$ when $x = x_0$? Estimate the
mean squared error of this predictor.
 A: For (a), note that 
$E(Y \vert x=x_0) = \alpha + \beta x_0$.  Thus, what do you think would be a reasonable estimate for $E(Y \vert x=x_0)$?  In regards to the MSE, recall that MSE can be decomposed into two terms, $Var$ and $Bias^2$.  What do you know about your estimators for $\alpha$ and $\beta$?  Are they unbiased?  As a result, is your estimator of the the mean of $Y$ biased or unbiased?  
In regards to the variance of your estimator, you might recall some useful properties of variances.  In general, for constants a and b,
$Var( aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X,Y)$
In regard to (b), what should be your predicted value of $Y$?  Note that an individual $Y$ includes an error term, but what should you predict for that error term?  Should the predicted $Y$ value be any different from your estimated mean of $Y$ when $x=x_0$?
In regard to the variance of your predicted value, the main difference between part a and b is that now you have to take into account the estimated variability around the regression line in addition to the variability encountered when estimating the regression line.  So 
$Var( \hat{\alpha} + \hat{\beta} x_0 + \epsilon) = Var( \hat{\alpha} + \hat{\beta}x_0 ) + Var(\epsilon)$
I hope that helps nudge you in the right direction toward solving your problem.
