Applied Statistics and Probability for Engineers

which would be compared to f0.05,6,123.00, and so H 0 : ^2 0 cannot be rejected.

TheP-value for this ratio is P0.2338. To test for no machine effect (H 0 : ^2 0), we

compute

which would be compared to f0.05,3,64.76. The P-value for this ratio is P0.3403.

Therefore, we conclude that machines do not significantly affect the breaking strength test

results. To test for no operator effect (H 0 : ^2 0), we compute

which would be compared to f0.05,2,65.14. The P-value for this ratio is P0.1904. We

conclude that the operators do not affect the breaking strength test results. The variance com-

ponents may be estimated using Equations S14-6 as follows:

All of the variance components , , and are small; in fact, they are not significantly

different from zero.

The Mixed Model

Now suppose that one of the factors, A, is fixed and the other, B, is random. This is called the

mixed modelanalysis of variance. The linear model is

Yijk ij 1  (^2) ijij k • (S14-7)
i1, 2,... , a
j1, 2,... , b
k1, 2,... , n
ˆ^2 ˆ^2  ˆ^2
ˆ^2 
8.11 5.94
6
0.36
ˆ^2 
13.17 5.94
8
0.90
ˆ^2 
5.94 3.75
2
1.10
ˆ^2 3.75
f 0 
MSA
MSAB

13.17
5.94
2.217
f 0 
MSB
MSAB

8.11
5.94
1.365
14-6
Table S14-2 Analysis of Variance for Example 14-2A
Source of Sum of Degrees of Mean
Variation Squares Freedom Square f 0 P-Value
Operators 26.33 2 13.17 2.217 0.1904
Machines 24.33 3 8.11 1.365 0.3403
Interaction 35.67 6 5.94 1.584 0.2338
Error 45.00 12 3.75
Total 131.33 23
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