For the simple linear regression model Y = ?0 + ?1 x + ?

Chapter 11, Problem 35E

(choose chapter or problem)

For the simple linear regression model \(Y=\beta_{0}+\beta_{1} x+\varepsilon\) with \(E(\varepsilon)=0\) and \(V(\varepsilon)=o^{2}\), use the expression for \(V\left(a_{0} \tilde{\beta_{0}}+a_{1} \widehat{\beta_{1}}\right)\) derived in this section to show that

\(V\left(\widehat{\beta_{0}}+\widehat{\beta_{1}} x\right)=\left[\frac{1}{n}+\frac{\left(x^{*}-\bar{x}\right)^{2}}{S_{m x}}\right] o^{2}\)

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

Equation transcription:

Text transcription:

Y=beta{0}+\beta{1} x+\varepsilon

E(\varepsilon)=0

V(\varepsilon)=o^{2}

V\(a{0} \tilde{\beta{0}}+a{1} \widehat{\beta{1}})

V(\widehat{\beta{0}}+\widehat{\beta{1}} x)=[\frac{1}{n}+\frac(x^{*}-\bar{x})^{2}}{S{m x}] o^{2}