Applied Statistics and Probability for Engineers

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

Graphical interpretation of the data is always useful. Box plots show the variability

of the observations withina treatment (factor level) and the variability betweentreatments.

We now discuss how the data from a single-factor randomized experiment can be analyzed

statistically.

13-2.2 The Analysis of Variance

Suppose we have adifferent levels of a single factor that we wish to compare. Sometimes,

each factor level is called a treatment,a very general term that can be traced to the early

applications of experimental design methodology in the agricultural sciences. The response

for each of the atreatments is a random variable. The observed data would appear as shown

in Table 13-2. An entry in Table 13-2, say yij, represents the jth observation taken under treat-

ment i. We initially consider the case in which there are an equal number of observations, n,

on each treatment.

We may describe the observations in Table 13-2 by the linear statistical model

(13-1)

where Yijis a random variable denoting the (ij)th observation, is a parameter common to all

treatments called the overall mean,iis a parameter associated with the ith treatment called

the ith treatment effect,and ijis a random error component. Notice that the model could

have been written as

where iiis the mean of the ith treatment. In this form of the model, we see

that each treatment defines a population that has mean i, consisting of the overall mean 

plus an effect ithat is due to that particular treatment. We will assume that the errors ij

are normally and independently distributed with mean zero and variance ^2. Therefore,

each treatment can be thought of as a normal population with mean iand variance ^2. See

Fig. 13-1(b).

Equation 13-1 is the underlying model for a single-factor experiment. Furthermore, since

we require that the observations are taken in random order and that the environment (often

called the experimental units) in which the treatments are used is as uniform as possible, this

experimental design is called a completely randomized design.

Yijiij e

i1, 2,p, a

j1, 2,p, n

Yijiij e

i1, 2,p, a

j1, 2,p, n

Table 13-2 Typical Data for a Single-Factor Experiment

Treatment Observations Totals Averages

1 y 11 y 12 p y 1 n y 1.

2 y 21 y 22 p y 2 n y 2.

aya 1 ya 2 p yan ya.

y.. y..

ya.

o o o ooo  o o

y 2.

y 1.

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