A scientist collects experimental data on the radius of a propellant grain (y) as a function of powder temperature, x1, extrusion rate, x2, and die temperature, x3. The data is presented in Table 3.26. (a). Consider the linear regression model yi = β ? 0 + β1(x1i − x¯1) + β2(x2i − x¯2) + β3(x3i − x¯3) + ²i . Write the vector y, the matrix X, and vector β in the model y = Xβ + ε.

Grain Radius Powder Temp (x1) Extrusion Rate (x2) Die Temp (x3) 82 150 12 220 92 190 12 220 114 150 24 220 124 150 12 250 111 190 24 220 129 190 12 250 157 150 24 250 164 190 24 250 (b). Write out the normal equation (X 0 X)b = X 0 y. Comment on what is special about the X 0 X matrix. What characteristic in this experiment do you suppose to produce this special form of X 0 X. (c). Estimate the coefficients in the multiple linear regression model. (d). Test the hypothesis H0 : Lβ1 = 0, H0 : β2 = 0 and make conclusion. (e). Compute 100(1−α)% confidence interval on E(y|x) at each of the locations of x1, x2, and x3 described by the data points. (f). Compute the HAT diagonals at eight data points and comment. (g). Compute the variance inflation factors of the coefficients b1, b2, and b3. Do you have any explanations as to why these measures of damage due to collinearity give the results that they do?