426 CHAPTER 12 MULTIPLE LINEAR REGRESSION(c) Compute the standard errors of the regression coefficients.
(d) Predict power consumption for a month in which
x 2 24 days, x 3 90%, and x 4 98 tons.
12-7. A study was performed on wear of a bearing yand its
relationship to x 1 oil viscosity and x 2 load. The follow-
ing data were obtained.x 1 75 F,(a) Fit a multiple linear regression model to these data.
(b) Estimate ^2 and the standard errors of the regression
coefficients.
(c) Use the model to predict wear when x 1 25 and x 2 1000.
(d) Fit a multiple linear regression model with an interaction
term to these data.
(e) Estimate ^2 and se( ) for this new model. How did these
quantities change. Does this tell you anything about the
value of adding the interaction term to the model?
(f ) Use the model in (d) to predict when x 1 25 and x 2 - Compare this prediction with the predicted value
from part (b) above.
12-8. The pull strength of a wire bond is an important charac-
teristic. The following table gives information on pull strength
(y), die height (x 1 ), post height (x 2 ), loop height (x 3 ), wire length
(x 4 ), bond width on the die (x 5 ), and bond width on the post (x 6 ).
ˆj(a) Fit a multiple linear regression model using x 2 , x 3 , x 4 , and
x 5 as the regressors.
(b) Estimate ^2.
(c) Find the se( ). How precisely are the regression coeffi-
cients estimated, in your opinion?
(d) Use the model from part (a) to predict pull strength when
x 2 = 20, x 3 = 30, x 4 = 90, and x 5 = 2.0.
12-9. An engineer at a semiconductor company wants to
model the relationship between the device HFE (y) and three
parameters: Emitter-RS (x 1 ), Base-RS (x 2 ), and Emitter-to-Base
RS (x 3 ). The data are shown in the following table.ˆjyx 1 x 2 x 3 x 4 x 5 x 6
8.0 5.2 19.6 29.6 94.9 2.1 2.3
8.3 5.2 19.8 32.4 89.7 2.1 1.8
8.5 5.8 19.6 31.0 96.2 2.0 2.0
8.8 6.4 19.4 32.4 95.6 2.2 2.1
9.0 5.8 18.6 28.6 86.5 2.0 1.8
9.3 5.2 18.8 30.6 84.5 2.1 2.1
9.3 5.6 20.4 32.4 88.8 2.2 1.9
9.5 6.0 19.0 32.6 85.7 2.1 1.9
9.8 5.2 20.8 32.2 93.6 2.3 2.1
10.0 5.8 19.9 31.8 86.0 2.1 1.8
10.3 6.4 18.0 32.6 87.1 2.0 1.6
10.5 6.0 20.6 33.4 93.1 2.1 2.1
10.8 6.2 20.2 31.8 83.4 2.2 2.1
11.0 6.2 20.2 32.4 94.5 2.1 1.9
11.3 6.2 19.2 31.4 83.4 1.9 1.8
11.5 5.6 17.0 33.2 85.2 2.1 2.1
11.8 6.0 19.8 35.4 84.1 2.0 1.8
12.3 5.8 18.8 34.0 86.9 2.1 1.8
12.5 5.6 18.6 34.2 83.0 1.9 2.0x 1 x 2 x 3 y
Emitter-RS Base-RS E-B-RS HFE-1M-5V
14.620 226.00 7.000 128.40
15.630 220.00 3.375 52.62
14.620 217.40 6.375 113.90
15.000 220.00 6.000 98.01
14.500 226.50 7.625 139.90
15.250 224.10 6.000 102.60
16.120 220.50 3.375 48.14
15.130 223.50 6.125 109.60
15.500 217.60 5.000 82.68
15.130 228.50 6.625 112.60
15.500 230.20 5.750 97.52
16.120 226.50 3.750 59.06
15.130 226.60 6.125 111.80
15.630 225.60 5.375 89.09
15.380 229.70 5.875 101.00
14.380 234.00 8.875 171.90
15.500 230.00 4.000 66.80
14.250 224.30 8.000 157.10
14.500 240.50 10.870 208.40
14.620 223.70 7.375 133.40yx 1 x 2
293 1.6 851
230 15.5 816
172 22.0 1058
91 43.0 1201
113 33.0 1357
125 40.0 1115(a) Fit a multiple linear regression model to the data.
(b) Estimate ^2.
(c) Find the standard errors se
(d) Predict HFE when x 1 14.5, x 2 220, and x 3 5.0.
12-10. Heat treating is often used to carburize metal parts,
such as gears. The thickness of the carburized layer is consid-
ered a crucial feature of the gear and contributes to the overall
reliability of the part. Because of the critical nature of this fea-
ture, two different lab tests are performed on each furnace
load. One test is run on a sample pin that accompanies each
load. The other test is a destructive test, where an actual part is
cross-sectioned. This test involves running a carbon analysis
on the surface of both the gear pitch (top of the gear tooth) and
the gear root (between the gear teeth). Table 12-7 shows the
results of the pitch carbon analysis test for 32 parts.
The regressors are furnace temperature (TEMP), carbon
concentration and duration of the carburizing cycle1 ˆj 2.c12 .qxd 6/4/02 2:30 PM Page 426 RK UL 6 RK UL 6:Desktop Folder:montgo: