A Designed Experiment for Linear Regression. You wish to fit a simple linear regression model over the region \(-1 \leq x \leq 1\) using \(n=10\) observa-tions. Four experimental designs are under consideration: (i) 5 observations at \(x=-1\) and 5 observations at \(x=+1\), (ii) 4 observations at \(x=-1,2\) observations at \(x=0\), and 4 observations at \(x=+1\), (iii) 2 observations at \(x=-1,-\frac{1}{2}, 0,+\frac{1}{2}\), and +1 , and (iv) 1 observation at \(x=-1,-0.8,-0.6,-0.4\), \(-0.2,+0.2,+0.4,+0.6,+0.8\), and +1 . For each of these designs, find the number of degrees of freedom available for evaluating pure error and testing lack of fit, the standard error of the slope (up to a constant \(\sigma\) ), and the value of the determinant of \(\mathbf{X}^{\prime} \mathbf{X}\). Based on these analyses, which design would you select?