dealt with howto form a confidence interval for the value

Chapter , Problem 14

(choose chapter or problem)

Problem 13 dealt with how to form a confidence interval for the value of a line at a point \(x_0\). Suppose that instead we want to predict the value of a new observation, \(Y_0\), at \(x_0\)

                                                          \(Y_0=\beta_0+\beta_1 x_0+e_0\)

by the estimate

                                                          \(\hat{Y}_0=\hat{\beta}_0+\hat{\beta}_1 x_0\)

a. Find an expression for the variance of \(\hat{Y}_0-Y_0\), and compare it to the expression for the variance of \(\hat{\mu}_0\) obtained in part (a) of Problem 13. Assume that \(e_0\) is independent of the original observations and has the variance \(\sigma^2\).

b. Assuming that \(e_0\) is normally distributed, find the distribution of \(\hat{Y}_0-Y_0\). Use this result to find an interval I such that \(P\left(Y_0 \in I\right)=1-\alpha\). This interval is called a \(100(1-\alpha) \%\) prediction interval.

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