Applied Statistics and Probability for Engineers

homogeneous conditions. Alternatively, we can view the paired t-test as a method for re-

ducing the background noise in the experiment by blocking out a nuisance factoreffect.

The block is the nuisance factor, and in this case, the nuisance factor is the actual experi-

mental unit—the steel girder specimens used in the experiment.

The randomized block design is an extension of the paired t-test to situations where

the factor of interest has more than two levels; that is, more than two treatments must be

compared. For example, suppose that three methods could be used to evaluate the strength

readings on steel plate girders. We may think of these as three treatments, say t 1 , t 2 , and t 3.

If we use four girders as the experimental units, a randomized complete block design

would appear as shown in Fig. 13-8. The design is called a randomized complete block

design because each block is large enough to hold all the treatments and because the actual

assignment of each of the three treatments within each block is done randomly. Once the

experiment has been conducted, the data are recorded in a table, such as is shown in

Table 13-9. The observations in this table, say yij, represent the response obtained when

method iis used on girder j.

The general procedure for a randomized complete block design consists of selecting b

blocks and running a complete replicate of the experiment in each block. The data that re-

sult from running a randomized complete block design for investigating a single factor

with alevels and bblocks are shown in Table 13-10. There will be aobservations (one per

factor level) in each block, and the order in which these observations are run is randomly

assigned within the block.

We will now describe the statistical analysis for a randomized complete block design.

Suppose that a single factor with alevels is of interest and that the experiment is run in b

blocks. The observations may be represented by the linear statistical model

Yijijij e (13-26)

i1, 2,p, a

j1, 2,p, b

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

Block 1

t 1

t 2

t 3

Block 2

t 1

t 2

t 3

Block 3

t 1

t 2

t 3

Block 4

t 1

t 2

t 3

Figure 13-8 A randomized complete

block design.

Table 13-9 A Randomized Complete Block Design

Block (Girder)

1234

1 y 11 y 12 y 13 y 14

2 y 21 y 22 y 23 y 24

3 y 31 y 32 y 33 y 34

Treatments

(Method)

Table 13-10 A Randomized Complete Block Design withaTreatments and bBlocks

Blocks

Treatments 1 2 p b Totals Averages

1 y 11 y 12 p y 1 b

2 y 21 y 22 p y 2 b

aya 1 ya 2 p yab ya.

Totals p y..

Averages y. 1 y. 2 p y.b y..

y. 1 y. 2 y.b

ya.

o o o o o o

y 2. y 2.

y 1. y 1.

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