Introduction to Probability and Statistics for Engineers and Scientists

468 Chapter 10:Analysis of Variance

Thus we have the following significance levelαtest of

H 0 r:allαi= 0

versus

H 1 r:at least oneαi= 0

Namely,

reject H 0 r if

SSr/(m−1)

SSe/nm(l−1)

>Fm−1,nm(l−1),α

do not reject H 0 r otherwise

Alternatively, if

SSr/(m−1)

SSe/nm(l−1)

=v, then

p-value=P{Fm−1,nm(l−1)>v}

Because an analogous result can be shown to hold when testingH 0 :allβj=0, we obtain
the ANOVA information shown in Table 10.4.
Note that all of the preceding tests call for rejection only when their relatedF-statistic
is large. The reason that only large (and not small) values call for rejection of the null
hypothesis is that the numerator of theF-statistic will tend to be larger whenH 0 is not
true than when it is, whereas the distribution of the denominator will be the same whether
or notH 0 is true.
Program 10.6 computes the values of theF-statistics and their associatedp-values.

EXAMPLE 10.6a The life of a particular type of generator is thought to be influenced by
the material used in its construction and also by the temperature at the location where it
is utilized. The following table represents lifetime data on 24 generators made from three
different types of materials and utilized at two different temperatures. Do the data indicate
that the material and the temperature do indeed affect the lifetime of a generator? Is there
evidence of an interaction effect?

Temperature

Material 10 ◦C18◦C

1 135, 150 50, 55

176, 85 64, 38

2 150, 162 76, 88

171, 120 91, 57

3 138, 111 68, 60

140, 106 74, 51

SOLUTION Run Program 10.6 (see Figures 10.3 and 10.4). ■