Consider the simple linear regression model Suppose that

Chapter 11, Problem 21E

(choose chapter or problem)

Consider the simple linear regression model \(Y=\beta_{0}+\beta_{1} x+\epsilon\). Suppose that the analyst wants to use \(z=x-\bar{x}\) as the regressor variable.

(a) Using the data in Exercise 11-13, construct one scatter plot of the \(\left(x_{i}, y_{i}\right)\) points and then another of the \(\left(z_{i}=x_{i}-\bar{x}, y_{i}\right)\) points. Use the two plots to intuitively explain how the two models, \(Y=\beta_{0}+\beta_{1} x+\epsilon \text { and } Y=\beta_{0}^{*}+\beta_{1}^{*} z+\epsilon\), are related.

(b) Find the least squares estimates of \(\beta_{0}^{*} \text { and } \beta_{i}^{*}\) in the model \(Y=\beta_{0}^{*}+\beta_{1}^{*} z+\epsilon\). How do they relate to the least squares estimates \(\hat{\beta}_{0} \text { and } \hat{\beta}_{1}\)?