Applied Statistics and Probability for Engineers

The ANOVA is summarized in Table 13-13. Since f 0 75.13 f0.01,3,125.95 (the P-value

is 4.79 

10 ^8 ), we conclude that there is a significant difference in the chemical types so far

as their effect on strength is concerned.

When Is Blocking Necessary?

Suppose an experiment is conducted as a randomized block design, and blocking was not

really necessary. There are abobservations and (a1)(b1) degrees of freedom for error.

If the experiment had been run as a completely randomized single-factor design with brepli-

cates, we would have had a(b1) degrees of freedom for error. Therefore, blocking has cost

a(b1)(a1)(b1)b1 degrees of freedom for error. Thus, since the loss in

error degrees of freedom is usually small, if there is a reasonable chance that block effects may

be important, the experimenter should use the randomized block design.

For example, consider the experiment described in Example 13-5 as a single-factor experi-

ment with no blocking. We would then have 16 degrees of freedom for error. In the randomized

block design, there are 12 degrees of freedom for error. Therefore, blocking has cost only 4

degrees of freedom, which is a very small loss considering the possible gain in information that

would be achieved if block effects are really important. The block effect in Example 13-5 is

large, and if we had not blocked, SSBlockswould have been included in the error sum of squares

for the completely randomized analysis. This would have resulted in a much larger MSE, making

it more difficult to detect treatment differences. As a general rule, when in doubt as to the

importance of block effects, the experimenter should block and gamble that the block effect does

exist. If the experimenter is wrong, the slight loss in the degrees of freedom for error will have a

negligible effect, unless the number of degrees of freedom is very small.

Computer Solution

Table 13-14 presents the computer output from Minitab for the randomized complete block

design in Example 13-5. We used the analysis of variance menu for balanced designs to solve

this problem. The results agree closely with the hand calculations from Table 13-13. Notice

that Minitab computes an F-statistic for the blocks (the fabric samples). The validity of this ra-

tio as a test statistic for the null hypothesis of no block effects is doubtful because the blocks

represent a restriction on randomization;that is, we have only randomized within the

blocks. If the blocks are not chosen at random, or if they are not run in random order, the

25.696.6918.040.96

SSESSTSSBlocksSSTreatments



1 9.2 22  1 10.1 22  1 3.5 22  1 8.8 22  1 7.6 22

4



1 39.2 22

20

6.69

SSBlocks a

5

j 1

y.^2 j

a

y..^2

ab

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

Table 13-13 Analysis of Variance for the Randomized Complete Block Experiment

Source of Degrees of

Variation Sum of Squares Freedom Mean Square f 0 P-value

Chemical types

(treatments) 18.04 3 6.01 75.13 4.79 E-8

Fabric samples

(blocks) 6.69 4 1.67

Error 0.96 12 0.08

Total 25.69 19

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