Applied Statistics and Probability for Engineers

12-2 HYPOTHESIS TESTS IN MULTIPLE LINEAR REGRESSION 435

EXAMPLE 12-5 Consider the wire bond pull strength data in Example 12-1. We will investigate the contribution of

the variable x 2 (die height) to the model using the partial F-test approach. That is, we wish to test

To test this hypothesis, we need the extra sum of squares due to  2 , or

In Example 12-3 we have calculated

and from Example 11-8, where we fit the model Y 0  1 x 1 , we can calculate

Therefore,

This is the increase in the regression sum of squares due to adding x 2 to a model already con-

taining x 1. To test H 0 :  2 0, calculate the test statistic

Note that the MSEfrom the full model, using both x 1 and x 2 , is used in the denominator of the

test statistic. Since f0.05,1,224.30, we reject H 0 :  2 0 and conclude that the regressor die

height (x 2 ) contributes significantly to the model.

Table 12-4 shows the Minitab regression output for the wire bond pull strength data. Just

below the analysis of variance summary in this table the quantity labeled “Seq SS” shows the

sum of squares obtained by fitting x 1 alone (5885.9) and the sum of squares obtained by fitting

x 2 after x 1. Notationally, these are referred to above as and.

Since the partial F-test in the above example involves a single variable, it is equivalent to

the t-test. To see this, recall from Example 12-5 that the t-test on H 0 :  2 0 resulted in the test

statistic t 0 4.4767. Furthermore, the square of a t-random variable with degrees of free-

dom is an F-random variable with one and degrees of freedom, and we note that 

(4.4767)^2 20.04f 0.

12-2.3 More About the Extra Sum of Squares Method (CD Only)

t^20

SSR 1  1 0  02 SSR 1  2 0  1 , 02

f 0 

SSR 1  2 0  1 , 02  1

MSE



104.9191 1

5.2352

20.04

104.9191 1 one degree of freedom 2

SSR 1  2 0  1 , 02 5990.77125885.8521

5885.8521 1 one degree of freedom 2

SSR 1  1 0  02 ˆ 1 Sxy 1 2.9027 21 2027.7132 2

SSR 1  1 , 2 0  02 ˆ¿X¿y

aa

n

i 1

yib

n 5990.7712^1 two degrees of freedom^2

SSR 1  1 , 2 0  02 SSR 1  1 0  02

SSR 1  2 0  1 , 02 SSR 1  1 , 2 , 02 SSR 1  1 , 02

H 1 :  2

0

H 0 :  2  0

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