Applied Statistics and Probability for Engineers

14-5

EXAMPLE S14-1 Two factors that may influence the breaking strength of cloth are being studied. Four test

machines and three operators are chosen at random, and an experiment is run using cloth from

the same production segment. The data are shown in Table S14-1, and the analysis of variance

is in Table S14-2. Notice that the first four columns in Table S14-2. are computed as in a stan-

dard (fixed-effects model) analysis. The test statistics are computed using Equations (S14-3)

through S14-5. We will use 0.05. The test statistic for the no-interaction hypothesis

H 0 : ^2 0 is

f 0 

MSAB

MSE



5.94

3.75

1.584

which is distributed as Fa 1,(a 1)(b 1), and for testingH 0 : ^2 0 the test statistic is

F 0  (S14-5)

MSB

MSAB

which is distributed as Fb 1,(a 1)(b 1). These are all upper-tail, one-tail tests. Thus, the null

hypotheses above would be rejected at the level of significance if the calculated value of f 0

exceeds the upper percentage point of the F-distribution. Notice that these test statistics are

not the same as those used if both factors Aand Bare fixed. The expected mean squares are

always used as a guide to test statistic construction.

The variance components may be estimated by equating the observed mean squares to

their expected values and solving for the variance components. This yields

(S14-6)

ˆ^2 

MSA MSAB

bn

ˆ^2 

MSB MSAB

an

ˆ^2 

MSAB MSE

n

ˆ^2 MSE

Table S14-1 Breaking Strength Data for Example S14-1

Machine

Operator 1 2 3 4

A 113 113 111 113

112 118 111 119

B 111 110 111 114

112 111 109 112

C 109 112 114 111

111 115 112 112

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