464 CHAPTER 12 MULTIPLE LINEAR REGRESSION(b) Use the t-statistic to test H 0 : j 0 versus H 1 : j 0 for
each variable in the model. If 0.01, what conclusions
can you draw?
(c) Plot the residuals versus and versus. Comment on
these plots. How do they compare with their counterparts
obtained in Exercise 12-64 parts (f ) and (g)?
12-66. The transient points of an electronic inverter are
influenced by many factors. Table 12-20 gives data on the
transient point (y, in volts) of PMOS-NMOS inverters and five
candidate regressors:, x 1 width of the NMOS device, x 2
length of the NMOS device, x 3 width of the PMOS device,
x 4 length of the PMOS device, and x 5 temperature (°C).
(a) Fit the multiple linear regression model to these data. Test
for significance of regression using 0.01. Find the
P-value for this test and use it to draw your conclusions.
(b) Test the contribution of each variable to the model using
the t-test with 0.05. What are your conclusions?
(c) Delete x 5 from the model. Test the new model for signifi-
cance of regression. Also test the relative contribution of
each regressor to the new model with the t-test. Using
0.05, what are your conclusions?yˆ* x* 3(d) Notice that the MSEfor the model in part (c) is smaller
than the MSEfor the full model in part (a). Explain why
this has occurred.
(e) Calculate the studentized residuals. Do any of these seem
unusually large?
(f ) Suppose that you learn that the second observation was in-
correctly recorded. Delete this observation and refit the
model using x 1 , x 2 , x 3 , and x 4 as the regressors. Notice that
the R^2 for this model is considerably higher than the R^2 for
either of the models fitted previously. Explain why the R^2
for this model has increased.
(g) Test the model from part (f ) for significance of regression
using 0.05. Also investigate the contribution of each
regressor to the model using the t-test with 0.05.
What conclusions can you draw?
(h) Plot the residuals from the model in part (f ) versus and
versus each of the regressors x 1 , x 2 , x 3 , and x 4. Comment
on the plots.
12-67. Consider the inverter data in Exercise 12-66. Delete
observation 2 from the original data. Define new variables as fol-
lows:
and
(a) Fit a regression model using these transformed regressors
(do not use x 5 ).
(b) Test the model for significance of regression using
0.05. Use the t-test to investigate the contribution of
each variable to the model ( 0.05). What are your
conclusions?
(c) Plot the residuals versus and versus each of the trans-
formed regressors. Comment on the plots.
12-68. Following are data on ygreen liquor (g/l) and x
paper machine speed (feet per minute) from a Kraft paper
machine. (The data were read from a graph in an article in the
Tappi Journal,March 1986.)yˆ*x* 4 1 x 4.yln y, x 1 (^1) 1 x 1 , x 2 1 x 2 , x 3 (^1) 1 x 3 ,
yˆ
(a) Fit the model using least
squares.
(b) Test for significance of regression using 0.05. What
are your conclusions?
(c) Test the contribution of the quadratic term to the model,
over the contribution of the linear term, using an F-statistic.
If 0.05, what conclusion can you draw?
(d) Plot the residuals from the model in part (a) versus.
Does the plot reveal any inadequacies?
(e) Construct a normal probability plot of the residuals.
Comment on the normality assumption.
yˆ
Y 0 1 x 2 x^2
Observation
Number x 1 x 2 x 3 x 4 x 5 y
1 3 3 3 3 0 0.787
2 8 30 8 8 0 0.293
3 3 6 6 6 0 1.710
4 4 4 4 12 0 0.203
5 8 7 6 5 0 0.806
6 10 20 5 5 0 4.713
7 8 6 3 3 25 0.607
8 6 24 4 4 25 9.107
9 4 10 12 4 25 9.210
10 16 12 8 4 25 1.365
11 3 10 8 8 25 4.554
12 8 3 3 3 25 0.293
13 3 6 3 3 50 2.252
14 3 8 8 3 50 9.167
15 4 8 4 8 50 0.694
16 5 2 2 2 50 0.379
17 2 2 2 3 50 0.485
18 10 15 3 3 50 3.345
19 15 6 2 3 50 0.208
20 15 6 2 3 75 0.201
21 10 4 3 3 75 0.329
22 3 8 2 2 75 4.966
23 6 6 6 4 75 1.362
24 2 3 8 6 75 1.515
25 3 3 8 8 75 0.751
y 16.0 15.8 15.6 15.5 14.8
x 1700 1720 1730 1740 1750
y 14.0 13.5 13.0 12.0 11.0
x 1760 1770 1780 1790 1795
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