Chapter 13.4, Problem 51E

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 5 surface roughness (μm) when a semisynthetic cutting fluid was used:

x1	x2	x3	y	e*
320	.10	15	2.27	−1.32
320	.12	20	4.14	1.08
320	.14	25	4.69	0.26
420	.10	20	1.92	−0.40
420	.12	25	2.63	−0.79
420	.14	15	4.34	0.99
520	.10	25	2.03	1.64
520	.12	15	2.34	0.03
520	.14	20	2.67	−1.52

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):

Predictor Constant	Coef	SE Coef	T	P
	0 . 099	1. 871	0 . 05	0 . 960
x1	−0.006767	0.002231	−3 . 03	0 . 029
x2	45 . 67	11.16	4 . 09	0 . 009
X3	0 . 01333	0 . 04463	0 .30	0 . 777
S = 0.546589 R-Sq = 83.9% R-Sq (adj ) = 74.2				2%

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):

Predictor Constant	Coef	SE Coef	T	P
	0.365	1.514	0 .24	0 . 817
x1	−0 . 006767	0.002055	−3 .29	0 . 017
x2	45.67	10 .28	4 .44	0 . 004
S = 0. 503400 R-Sq =83.6% R-Sq(adj) =78.1%

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.