Applied Statistics and Probability for Engineers

13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 475

Note that the error mean square

is an unbiased estimator of ^2 regardless of whether or not H 0 is true. We can also show that

MSTreatmentsand MSEare independent. Consequently, we can show that if the null hypothesis H 0

is true, the ratio

MSESSE
3 a 1 n 124

F 0  (13-7)

SSTreatments 

1 a 12

SSE 

3 a 1 n 124



MSTreatments

MSE

The sums of squares computing formulas for the ANOVA with equal sample sizes in

each treatment are

(13-8)

and

(13-9)

The error sum of squares is obtained by subtraction as

SSESSTSSTreatments (13-10)

SS (^) Treatments a
a
i 1
yi^2
n
y..^2
N
SS (^) T a
a
i 1
(^) a
n
j 1
yij^2 
y..^2
N
Definition
has an F-distribution with a1 and a(n1) degrees of freedom. Furthermore, from the ex-
pected mean squares, we know that MSEis an unbiased estimator of ^2. Also, under the null
hypothesis, MSTreatmentsis an unbiased estimator of ^2. However, if the null hypothesis is false,
the expected value of MSTreatmentsis greater than ^2. Therefore, under the alternative hypothe-
sis, the expected value of the numerator of the test statistic (Equation 13-7) is greater than the
expected value of the denominator. Consequently, we should reject H 0 if the statistic is large.
This implies an upper-tail, one-tail critical region. Therefore, we would reject H 0 if
where f 0 is the computed value of F 0 from Equation 13-7.
Efficient computational formulas for the sums of squares may be obtained by expanding
and simplifying the definitions of SSTreatmentsand SST. This yields the following results.
f 0 f ,a1,a 1 n 12
The computations for this test procedure are usually summarized in tabular form as shown in
Table 13-3. This is called an analysis of variance(orANOVA) table.
Table 13-3 The Analysis of Variance for a Single-Factor Experiment, Fixed-Effects Model
Source of Degrees of
Variation Sum of Squares Freedom Mean Square F 0
Treatments SSTreatments a 1 MSTreatments
Error SSE a(n1) MSE
Total SST an 1
MSTreatments
MSE
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