Applied Statistics and Probability for Engineers

13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 483

type II error () for various sample sizes against a measure of the difference in means that it

is important to detect. Thus, if the experimenter knows the magnitude of the difference in

means that is of potential importance, the operating characteristic curves can be used to deter-

mine how many replicates are required to achieve adequate sensitivity.

The power of the ANOVA test is

(13-17)

To evaluate this probability statement, we need to know the distribution of the test statistic F 0

if the null hypothesis is false. It can be shown that, if H 0 is false, the statistic F 0 

MSTreatmentsMSEis distributed as a noncentral Frandom variable,with a1 and a(n1)

degrees of freedom and a noncentrality parameter . If 0, the noncentral F-distribution

becomes the usual or central F-distribution.

Operating characteristic curves are used to evaluate defined in Equation 13-17. These

curves plot against a parameter , where

(13-18)

The parameter ^2 is (apart from n) the noncentrality parameter . Curves are available for

0.05 and 0.01 and for several values of the number of degrees of freedom for nu-

merator (denoted v 1 ) and denominator (denoted v 2 ). Figure 13-6 gives representative O.C.

curves, one for a4 (v 1 3) and one for a5 (v 1 4) treatments. Notice that for each

value of athere are curves for 0.05 and 0.01. O.C. curves for other values of aare

in Section 13-2.7 on the CD.

In using the operating curves, we must define the difference in means that we wish to

detect in terms of. Also, the error variance ^2 is usually unknown. In such cases, we

must choose ratios of that we wish to detect. Alternatively, if an estimate of ^2

is available, one may replace ^2 with this estimate. For example, if we were interested in

the sensitivity of an experiment that has already been performed, we might use MSEas the

estimate of ^2.

EXAMPLE 13-3 Suppose that five means are being compared in a completely randomized experiment with

0.01. The experimenter would like to know how many replicates to run if it is impor-

tant to reject H 0 with probability at least 0.90 if. The parameter ^2 is, in

this case,

and for the operating characteristic curve with v 1 a 1  5  1 4, and v 2 a(n1)

5(n1) error degrees of freedom refer to the lower curve in Figure 13-6. As a first guess, try

n4 replicates. This yields ^2 4, 2, and v 2 5(3)15 error degrees of freedom.

Consequently, from Figure 13-6, we find that   0.38. Therefore, the power of the test is

approximately 1 1 0.380.62, which is less than the required 0.90, and so we

^2 

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5

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5

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P 5 F 0 f ,a1,a 1 n 12 0 H 0 is false 6
1 P 5 Reject H 0 0 H 0 is false 6
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