However, since the (^) ij’s are independent random variables with mean zero and variance ^2 , we
find that
Therefore
As a result,
Now if the null hypothesis of equal treatment means is true, each iis equal to zero and
If the alternative hypothesis is true,
A similar approach will show that
so that
O.C. Curves
In the text, we give O.C. curves for the fixed effects ANOVA for the case of a4 and
a5 treatments. A collection of additional curves for a2, 3, 6, 7, 8, and 9 are on pages
13-8 through 13-10. In using the curves, remember that v 1 the number of numerator
degrees of freedom and v 2 the number of denominator degrees of freedom. The sample
size calculation routine in Minitab will also determine sample sizes for the single-factor
ANOVA.
E 1 MSE 2 E a
SSE
a 1 n 12
b^2
E 1 SSE 2 a 1 n 12 ^2
E a
SSTreatments
a 1
b^2
n (^) a
a
i 1
^2 i
a 1
E a
SSTreatments
a 1
b^2
^2
n (^) a
a
i 1
^2 i
a 1
1
a 1
E 1 SSTreatments 2
E 1 MSTreatments 2 E a
SSTreatments
a 1
b
1 a 12 ^2 n (^) a
a
i 1
^2 i
E 1 SSTreatments 2 na
a
i 1
^2 ia^2 ^2
E (^1 2) i. 2
^2
n,^ E^1
(^2) .. 2
2
an, and E^1 i.^2 ^0
13-7
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