Applied Statistics and Probability for Engineers

For the randomized block design, the relevant mean squares are

The expected values of these mean squares can be shown to be as follows:

MSE

SSE

1 a 121 b 12

MSBlocks

SSBlocks

b 1

MSTreatments

SSTreatments

a 1

494 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE

E 1 MSE 2 ^2

E 1 MSBlocks 2 ^2 

aa

b

j 1

^2 j

b 1

E 1 MSTreatments 2 ^2 

ba

a

i 1

^2 i

a 1

The computing formulas for the sums of squares in the analysis of variance for a ran-

domized complete block design are

(13-29)

(13-30)

(13-31)

and

SSESSTSSTreatmentsSSBlocks (13-32)

SSBlocks

1

a^ a

b

j 1

y

(^2).
j
y^2 ..
ab
SSTreatments
1
b
(^) a
a
i 1
y
2
i.
y^2 ..
ab
SST a
a
i 1
a
b
j 1
y
2
ij
y..^2
ab
Definition
Therefore, if the null hypothesis H 0 is true so that all treatment effects i0, MSTreatmentsis an
unbiased estimator of ^2 , while if H 0 is false, MSTreatmentsoverestimates ^2. The mean square
for error is always an unbiased estimate of ^2. To test the null hypothesis that the treatment ef-
fects are all zero, we use the ratio
(13-28)
which has an F-distribution with a1 and (a1)(b1) degrees of freedom if the null
hypothesis is true. We would reject the null hypothesis at the -level of significance if the
computed value of the test statistic in Equation 13-28 isf 0 f ,a 1 ,(a1)(b1).
In practice, we compute SST, SSTreatmentsand SSBlocksand then obtain the error sum of
squares SSEby subtraction. The appropriate computing formulas are as follows.
F 0 
MSTreatments
MSE
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