a. Show that the regression \(R^{2}\) in the regression of \(Y\) on \(X\) is the squared value of the sample correlation between \(X\) and \(Y\). That is, show that \(R^{2}=r_{X Y}^{2}\).

b. Show that the \(R^{2}\) from the regression of \(Y\) on \(X\) is the same as the \(R^{2}\) from the regression of \(X\) on \(Y\).

c. Show that \(\hat{\beta}_{1}=r_{X Y}\left(s_{Y} / s_{X}\right)\), where \(r_{X Y}\) is the sample correlation between \(X\) and \(Y\) and \(s_{X}\) and \(s_{Y}\) are the sample standard deviations of \(X\) and \(Y\).