Applied Statistics and Probability for Engineers

12-3 CONFIDENCE INTERVALS IN MULTIPLE LINEAR REGRESSION 437

12-3 CONFIDENCE INTERVALS IN MULTIPLE LINEAR REGRESSION

12-3.1 Confidence Intervalson Individual Regression Coefficients

In multiple regression models, it is often useful to construct confidence interval estimates for

the regression coefficients The development of a procedure for obtaining these confi-

dence intervals requires that the errors are normally and independently distributed with

mean zero and variance 2. This is the same assumption required in hypothesis testing.

Therefore, the observations {Yi} are normally and independently distributed with mean  0 

gkj 1 jxijand variance 2. Since the least squares estimator is a linear combination of the

observations, it follows that is normally distributed with mean vector and covariance

matrix. Then each of the statistics

(12-33)

has a tdistribution with npdegrees of freedom, where Cjjis the jjth element of the

matrix, and is the estimate of the error variance, obtained from Equation 12-16. This

leads to the following 100(1)% confidence interval for the regression coefficient

j, j0, 1, p, k.

ˆ^2

1 X¿X 2 ^1

T

ˆjj

2 ˆ^2 Cjj

j0, 1, p , k

21 X¿X 2 ^1

ˆ

ˆ

5 i 6

5 j 6.

Because is the standard error of the regression coefficient , we would also write the

CI formula as

EXAMPLE 12-6 We will construct a 95% confidence interval on the parameter  1 in the wire bond pull strength

problem. The point estimate of  1 is and the diagonal element of

corresponding to  1 is C 11 0.001671. The estimate of 2 is and t0.025,222.074.

Therefore, the 95% CI on  1 is computed from Equation 12-34 as

which reduces to

2.55029

 1

 2.93825

2.74427 1 2.074 221 5.2352 21 .001671 2 

 1

 2.74427 1 2.074 221 5.2352 21 .001671 2

ˆ^2 5.2352,

ˆ 1 2.74427 1 X¿X 2 ^1

ˆjt2,np se 1 ˆj 2 

j

ˆjt2,np se 1 ˆj 2.

2 ˆ^2 Cjj ˆj

(b) Use the t-test to evaluate the contribution of each

regressor to the model. Does it seem that all regressors are

necessary? Use 0.05.

(c) Fit a regression model relating the number of games won

to the number of points scored and the number of power

play goals. Does this seem to be a logical choice of

regressors, considering your answer to part (b)? Test this

new model for significance of regression and evaluate the

contribution of each regressor to the model using the

t-test. Use 0.05.

A 100(1) % confidence interval on the regression coefficient j, j0, 1, p,

kin the multiple linear regression model is given by

ˆjt2,np 2 ˆ^2 Cjj

j

ˆjt2,np 2 ˆ^2 Cjj (12-34)

Definition

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