Applied Statistics and Probability for Engineers

422 CHAPTER 12 MULTIPLE LINEAR REGRESSION

which is symmetric (C 10 C 01 ,C 20 C 02 , and C 21 C 12 ) because (XX)^1 is symmetric,

and we have

In general, the covariance matrix of is a (pp) symmetric matrix whose jjth element is the

variance of and whose i, jth element is the covariance between and that is,

The estimates of the variances of these regression coefficients are obtained by replacing

^2 with an estimate. When ^2 is replaced by it’s estimate , the square root of the estimated

variance of the jth regression coefficient is called the estimated standard errorof or

These standard errors are a useful measure of the precision of estimation

for the regression coefficients; small standard errors imply good precision.

Multiple regression computer programs usually display these standard errors. For

example, the Minitab output in Table 12-4 reports se 1 ˆ 02 1.060, se 1 ˆ 12 0.09352,and

se 1 ˆj 2  2 ˆ^2 Cjj.

ˆj

ˆ^2

cov 1 ˆ 2 ^21 X¿X 2 ^1 ^2 C

ˆj ˆi ˆj,

ˆ

cov 1 ˆi, ˆj 2 ^2 C ̨ij, ij

V 1 ˆ ̨j 2 ^2 C ̨jj, j0, 1, 2

The regression equation is

Strength2.26 2.74 Length 0.0125 Height

Predictor Coef SE Coef T P VIF

Constant ˆ 0 2.264 1.060 2.14 0.044

Length ˆ 1 2.74427 0.09352 29.34 0.000 1.2

Height ˆ 2 0.012528 0.002798 4.48 0.000 1.2

S2.288 R-Sq98.1% R-Sq (adj)97.9%

PRESS156.163 R-Sq (pred)97.44%

Analysis of Variance

Source DF SS MS F P

Regression 2 5990.8 2995.4 572.17 0.000

Residual Error 22 115.2 5.2

Total 24 6105.9

Source DF Seq SS

Length 1 5885.9

Height 1 104.9

Predicted Values for New Observations

New Obs Fit SE Fit 95.0% CI 95.0% PI

1 27.663 0.482 (26.663, 28.663) (22.814, 32.512)

Values of Predictors for New Observations

New Obs Length Height

1 8.00 275

ˆ^2

Table 12-4 Minitab Multiple Regression Output for the Wire Bond Pull Strength Data

Regression Analysis: Strength versus Length, Height

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