Applied Statistics and Probability for Engineers

13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 479

Since the CI includes zero, we would conclude that there is no difference in mean tensile

strength at these two particular hardwood levels.

The bottom portion of the computer output in Table 13-5 provides additional information con-

cerning which specific means are different. We will discuss this in more detail in Section 13-2.3.

An Unbalanced Experiment

In some single-factor experiments, the number of observations taken under each treatment

may be different. We then say that the design is unbalanced.In this situation, slight

modifications must be made in the sums of squares formulas. Let niobservations be taken

under treatment i(i1, 2,... , a), and let the total number of observations The

computational formulas for SSTand SSTreatmentsare as shown in the following definition.

Ngai 1 ni.

Choosing a balanced design has two important advantages. First, the ANOVA is relatively

insensitive to small departures from the assumption of equality of variances if the sample sizes

are equal. This is not the case for unequal sample sizes. Second, the power of the test is max-

imized if the samples are of equal size.

13-2.3 Multiple Comparisons Following the ANOVA

When the null hypothesis is rejected in the ANOVA, we know

that some of the treatment or factor level means are different. However, the ANOVA doesn’t

identify which means are different. Methods for investigating this issue are called multiple

comparisons methods.Many of these procedures are available. Here we describe a very

simple one, Fisher’s least significant difference(LSD) method.In Section 13-2.4 on the

CD, we describe three other procedures. Montgomery (2001) presents these and other methods

and provides a comparative discussion.

The Fisher LSD method compares all pairs of means with the null hypotheses H 0 : ij

(for all i j) using the t-statistic

Assuming a two-sided alternative hypothesis, the pair of means iand jwould be declared

significantly different if

0 yi.yj. 0 LSD

t 0 

yi.yj.

B

2 MSE

n

H 0 :  1  2 pa 0

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

niin each treatment are

(13-13)

(13-14)

and

SSESSTSSTreatments (13-15)

SS Treatments a

a

i 1

yi^2.

ni

y^2 ..

N

SST a

a

i 1

(^) a
ni
j 1
yij^2 
y^2 ..
N
Definition
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