Applied Statistics and Probability for Engineers

514 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

which has an F-distribution with (a1)(b1) and ab(n1) degrees of freedom if the null

hypothesis H 0 : ()ij0. This hypothesis is rejected at the level of significance if

f 0

 f ,(a1)(b1),ab(n1).

It is usually best to conduct the test for interaction first and then to evaluate the main

effects. If interaction is not significant, interpretation of the tests on the main effects is

straightforward. However, as noted in Section 14-4, when interaction is significant, the main

effects of the factors involved in the interaction may not have much practical interpretative

value. Knowledge of the interaction is usually more important than knowledge about the

main effects.

Computational formulas for the sums of squares are easily obtained.

The computations are usually displayed in an ANOVA table, such as Table 14-4.

EXAMPLE 14-1 Aircraft primer paints are applied to aluminum surfaces by two methods: dipping and spray-

ing. The purpose of the primer is to improve paint adhesion, and some parts can be primed

using either application method. The process engineering group responsible for this operation

is interested in learning whether three different primers differ in their adhesion properties.

Computing formulas for the sums of squares in a two-factor analysis of variance.

(14-5)

(14-6)

(14-7)

(14-8)

SSESSTSSABSSASSB (14-9)

SSAB a

a

i 1

(^) a
b
j 1
y^2 ij.
n 
y^2 ...
abn
SSASSB
SSB a
b
j 1
y.^2 j.
an
y^2 ...
abn
SSA a
a
i 1
y^2 i#..
bn

y^2 ...
abn
SST a
a
i 1
(^) a
b
j 1
(^) a
n
k 1
y^2 ijk
y^2 ...
abn
Definition
Table 14-4 ANOVA Table for a Two-Factor Factorial, Fixed-Effects Model
Source of Sum of Degrees of
Variation Squares Freedom Mean Square F 0
Atreatments SSA a 1
Btreatments SSB b 1
Interaction SSAB (a1)(b1)
Error SSE ab(n1)
Total SST abn 1 MSE
SSE
ab 1 n 12
MSAB
MSE
MSAB
SSAB
1 a 121 b 12
MSB
MSE
MSB
SSB
b 1
MSA
MSE
MSA
SSA
a 1
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