The article "Optimization of Surface Roughness in Drilling Using Vegetable-Based Cutting Oils Developed from Sunflower Oil" (Industrial Lubrication and Tribology, 2011: 271-276) gave the following data on x1 = spindle speed (rpm), x2 = feed rate (mm/rev), x3 = drilling depth (mm), and y = surface roughness (mm) when a semisynthetic cutting fluid was used:

a. Here is partial Minitab output from fitting the model with x1, x2, and x3 as predictors (authors of the cited article used Minitab for this purpose):

Does drilling depth provide useful information about roughness given that spindle speed and feed rate remain in the model?
b. Here is Minitab output from fitting the model with just x1 and x2 as predictors (the cited article made no mention of this model):

Carry out a test of model utility using α = .05.
c. Calculate and interpret a 95% CI for the population regression coefficient on x1.
d. The estimated standard deviation of the predicted Y when x1 = 400 and x2 = .125 is .180. Calculate a 95% CI for true average roughness under these circumstances.
e. The e* values that appear along with the data are from the regression of (b). Investigate model adequacy.

12 320 .10 15 2.27 320 12 20 4.14 320 14 25 4.69 420 420 .12 25 2.63-0.79 420 520 10 25 2.03 520 12 15 2.34 520 14 20 2.67 4 1.08 10 20 1.92-0.40 0.99 1.64 0.03 1.52 14 5 4.34 Predictor Constant , x2 X3 S:0.546589 Coef SE Coef 0.099 1.871 0.05 0.960 -0.006767 0.0022313.03 0.029 11.16 4.09 0.009 0.01333 0.04463 0.30 0.777 45.67 R-Sq 83.9% R-Sq (adj ):74.2% Coef SE Coef Constant 0.365 1.514 0.24 0.817 -3.29 0.017 10.28 4.44 0.004 -0.006767 0.002055 r2 45.67 S=0.503400 R-Sq=83.6% R-Sq (adj )=78.1%