Probability and Statistics for Engineering and the Sciences
9th Edition
ISBN: 9781305251809
Author: Jay L. Devore
Publisher: Cengage Learning
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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.
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Chapter 13 Solutions
Probability and Statistics for Engineering and the Sciences
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