Applied Statistics and Probability for Engineers

13-16

In the randomized block design, replace nby b, the number of blocks. Since ^2 is usually un-

known, we may either use a prior estimate or define the value of ^2 that we are interested in

detecting in terms of the ratio ^2 ^2.

EXAMPLE S13-3 Consider the situation described in Example S13-2. Since 0.05, a5, n6, and ^2 

^2 , we may find from Equation S13-4 as

From the operating characteristic curve with v 1 a     1 4, v 2 a(n 1)25 degrees

of freedom and 0.05, we find that

Therefore, the power is approximately 0.80. This agrees with the results obtained in

Example S13-2.

13-4.4 Randomized Complete Block Design with Random Factors (CD Only)

In the preceding sections, we have assumed that the treatments and blocks are fixed factors. In

many randomized complete block designs, these assumptions may be too restrictive. For

example, in the chemical type experiment, Example 13-5, we might like to view the fabric

samples as a random sample of material to which the chemicals may be applied so that the

conclusions from the experiment will extend to the entire population of material.

It turns out that, if either treatments or blocks (or both) are random effects, the F-test in

the analysis of variance is still formed as

This can be shown by using the methods presented previously to evaluate the expected mean

squares. If the treatments are random, the treatment effects iare considered to be normally

and independently distributed random variables with mean zero and variance ^2. The null

hypothesis of zero treatment effects is

When both treatments and blocks are random, the block effects jare also assumed to be

normally and independently distributed random variables with mean zero and variance ^2 . In

this case the expected values of the mean squares for treatments, blocks, and error are

The unbiased estimates of the variance components are

ˆ^2 

MSBlocks MSE

a

ˆ^2 

MSTreatments MSE

b

ˆ^2 MSE

E 1 MSE 2 ^2

E 1 MSBlocks 2 ^2 a^2 

E 1 MSTreatments 2 ^2 b^2 

H 1 : ^2  

0

H 0 : ^2  0

F 0 

MSTreatments

MSE

0.20

 11  6112 2.646

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