Chapter 11.6, Problem 35E

For the simple linear regression model Y = β₀ + β₁x + ε with E(ε) = 0 and V(ε) = σ², use the expression for $V(a_0{\hat{\beta }}_0+a_1{\hat{\beta }}_1)$ derived in this section to show that

$V\left({\hat{\beta }}_0+{\hat{\beta }}_1{x}^{*}\right)=\left[\frac{1}{n}+\frac{{(x^{*}-\bar{x})}^2}{S_{xx}}\right]{\sigma }^2.$

For what value of x^∗ does the confidence interval for E(Y) achieve its minimum length?