462 CHAPTER 12 MULTIPLE LINEAR REGRESSION(d) Compute the residuals from part (a) and use them to eval-
uate model adequacy.
12-48. When fitting polynomial regression models, we
often subtract from each xvalue to produce a “centered’’
regressor .This reduces the effects of dependencies
among the model terms and often leads to more accurate esti-
mates of the regression coefficients. Using the data from
Exercise 12-46, fit the model
Use the results to estimate the coefficients in the uncentered
model.
12-49. Suppose that we use a standardized variable
, where sxis the standard deviation of x, in
constructing a polynomial regression model. Using the data in
Exercise 12-46 and the standardized variable approach, fit the
model.
(a) What value of ydo you predict when?
(b) Estimate the regression coefficients in the unstandardized
model.
(c) What can you say about the relationship between SSEand
R^2 for the standardized and unstandardized models?
(d) Suppose that is used in the model along
with. Fit the model and comment on the relationship
between SSEand R^2 in the standardized model and the
unstandardized model.
12-50. The following data shown were collected during an
experiment to determine the change in thrust efficiency (y, in
percent) as the divergence angle of a rocket nozzle (x) changes:x¿y¿ 1 yy (^2) sy
Y 0 1 x 11 x^2
x 285 F
Y 0 1 x¿ 111 x¿ 22
x¿ 1 xx (^2) sx
Y 0 1 x 11 x^2
Y 0 1 x¿ 111 x¿ 22 .
x¿xx
x
(a) Fit a second-order model to the data.
(b) Test for significance of regression and lack of fit using
0.05.
(c) Test the hypothesis that 11 0, using 0.05.
(d) Plot the residuals and comment on model adequacy.
(e) Fit a cubic model, and test for the significance of the cubic
term using 0.05.
12-51. An article in the Journal of Pharmaceuticals
Sciences(Vol. 80, 1991, pp. 971–977) presents data on the ob-
served mole fraction solubility of a solute at a constant tem-
perature and the dispersion, dipolar, and hydrogen bonding
Hansen partial solubility parameters. The data are as shown in
the following table, where yis the negative logarithm of the
mole fraction solubility, x 1 is the dispersion partial solubility,
x 2 is the dipolar partial solubility, and x 3 is the hydrogen
bonding partial solubility.
(a) Fit the model
(b) Test for significance of regression using 0.05.
(c) Plot the residuals and comment on model adequacy.
12 x 1 x 2 13 x 1 x 3 23 x 2 x 3 11 x^21 22 x 22 33 x^23 .
Y 0 1 x 1 2 x 2 3 x 3
(d) Use the extra sum of squares method to test the contribu-
tion of the second-order terms using 0.05.
12-52. Consider the gasoline mileage data in Exercise 12-5.
(a) Discuss how you would model the information about the
type of transmission in the car.
(b) Fit a regression model to the gasoline mileage using
engine displacement, horsepower, and the type of trans-
mission in the car as the regressors.
(c) Is there evidence that the type of transmission affects
gasoline mileage performance?
12-53. Consider the tool life data in Example 12-12. Test
the hypothesis that two different regression models (with dif-
ferent slopes and intercepts) are required to adequately model
the data. Use indicator variables in answering this question.
12-54. Use the National Football League Team Performance
data in Exercise 12-4 to build regression models using the
following techniques:
(a) All possible regressions. Find the equations that minimize
MSEand that minimize Cp.
(b) Stepwise regression.
y 24.60 24.71 23.90 39.50 39.60 57.12
x 4.0 4.0 4.0 5.0 5.0 6.0
y 67.11 67.24 67.15 77.87 80.11 84.67
x 6.5 6.5 6.75 7.0 7.1 7.3
Observation
Number yx 1 x 2 x 3
1 0.22200 7.3 0.0 0.0
2 0.39500 8.7 0.0 0.3
3 0.42200 8.8 0.7 1.0
4 0.43700 8.1 4.0 0.2
5 0.42800 9.0 0.5 1.0
6 0.46700 8.7 1.5 2.8
7 0.44400 9.3 2.1 1.0
8 0.37800 7.6 5.1 3.4
9 0.49400 10.0 0.0 0.3
10 0.45600 8.4 3.7 4.1
11 0.45200 9.3 3.6 2.0
12 0.11200 7.7 2.8 7.1
13 0.43200 9.8 4.2 2.0
14 0.10100 7.3 2.5 6.8
15 0.23200 8.5 2.0 6.6
16 0.30600 9.5 2.5 5.0
17 0.09230 7.4 2.8 7.8
18 0.11600 7.8 2.8 7.7
19 0.07640 7.7 3.0 8.0
20 0.43900 10.3 1.7 4.2
21 0.09440 7.8 3.3 8.5
22 0.11700 7.1 3.9 6.6
23 0.07260 7.7 4.3 9.5
24 0.04120 7.4 6.0 10.9
25 0.25100 7.3 2.0 5.2
26 0.00002 7.6 7.8 20.7
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