Applied Statistics and Probability for Engineers

12-1 MULTIPLE LINEAR REGRESSION MODEL 411

3. Assess regression model adequacy


4. Test hypotheses and construct confidence intervals on the regression coefficients


5. Use the regression model to estimate the mean response and to make predictions and to construct
   confidence intervals and prediction intervals


6. Build regression models with polynomial terms


7. Use indicator variables to model categorical regressors


8. Use stepwise regression and other model building techniques to select the appropriate set of vari-
   ables for a regression model
   CD MATERIAL


9. Understand how ridge regression provides an effective way to estimate model parameters where
   there is multicollinearity.


10. Understand the basic concepts of fitting a nonlinear regression model.

Answers for many odd numbered exercises are at the end of the book. Answers to exercises whose

numbers are surrounded by a box can be accessed in the e-Text by clicking on the box. Complete

worked solutions to certain exercises are also available in the e-Text. These are indicated in the

Answers to Selected Exercises section by a box around the exercise number. Exercises are also

available for some of the text sections that appear on CD only. These exercises may be found within

the e-Text immediately following the section they accompany.

12-1 MULTIPLE LINEAR REGRESSION MODEL

12-1.1 Introduction

Many applications of regression analysis involve situations in which there are more than one

regressor variable. A regression model that contains more than one regressor variable is called

a multiple regression model.

As an example, suppose that the effective life of a cutting tool depends on the cutting speed

and the tool angle. A multiple regression model that might describe this relationship is

(12-1)

where Yrepresents the tool life, x 1 represents the cutting speed, x 2 represents the tool angle,

and is a random error term. This is a multiple linear regression modelwith two regressors.

The term linearis used because Equation 12-1 is a linear function of the unknown parameters

 0 ,  1 , and  2.

The regression model in Equation 12-1 describes a plane in the three-dimensional space

of Y, x 1 , and x 2. Figure 12-1(a) shows this plane for the regression model

where we have assumed that the expected value of the error term is zero; that is E()0.

The parameter  0 is the interceptof the plane. We sometimes call  1 and  2 partial regres-

sion coefficients,because  1 measures the expected change in Yper unit change in x 1 when

x 2 is held constant, and  2 measures the expected change in Yper unit change in x 2 when x 1

is held constant. Figure 12-1(b) shows a contour plotof the regression model— that is, lines

of constant E(Y) as a function of x 1 and x 2. Notice that the contour lines in this plot are

straight lines.

E 1 Y 2  50  10 x 1  7 x 2

Y 0  1 x 1  2 x 2 

c 12 .qxd 5/20/02 9:31 M Page 411 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: