A Designed Experiment for Linear Regression. You wish to fi t a simple linear regression model over the region − 1 ≤ x ≤ 1 using n = 10 observations.

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, − 1 2 , 0, + 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, fi nd the number of degrees of freedom available for evaluating pure error and testing lack of fi t, the standard error of the slope (up to a constant σ ), and the value of the determinant of X ′ X . Based on these analyses, which design would you select?