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A manufacturer of paper used for making grocery bags is interested in improving the tensile strength of the product. Product engineering thinks that tensile strength is a function of the hardwood concentration in the pulp and that the range of hardwood concentrations of practical interest is between 5 and 20%. A team of engineers responsible for the study decides to investigate four levels of hardwood concentration: 5%, 10%, 15%, and 20%. They decide to make up six test specimens at each concentration level, using a pilot plant. All 24 specimens are tested on a laboratory tensile tester, in random order. The data from this experiment are shown in Table 13-1.
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Table 13-1 Tensile Strength of Paper (psi) Hardwood Concentration (%) 5 10 15 20 Observations 3 4 15 11 13 18 19 17 22 23
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This is an example of a completely randomized single-factor experiment with four levels of the factor. The levels of the factor are sometimes called treatments, and each treatment has six observations or replicates. The role of randomization in this experiment is extremely important. By randomizing the order of the 24 runs, the effect of any nuisance variable that may in uence the observed tensile strength is approximately balanced out. For example, suppose that there is a warm-up effect on the tensile testing machine; that is, the longer the machine is on, the greater the observed tensile strength. If all 24 runs are made in order of increasing hardwood concentration (that is, all six 5% concentration specimens are tested rst, followed by all six 10% concentration specimens, etc.), any observed differences in tensile strength could also be due to the warm-up effect. It is important to graphically analyze the data from a designed experiment. Figure 13-1(a) presents box plots of tensile strength at the four hardwood concentration levels. This gure indicates that changing the hardwood concentration has an effect on tensile strength; specifically, higher hardwood concentrations produce higher observed tensile strength. Furthermore, the distribution of tensile strength at a particular hardwood level is reasonably symmetric, and the variability in tensile strength does not change dramatically as the hardwood concentration changes.
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Tensile strength (psi)
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10 15 20 Hardwood concentration (%) (a)
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Figure 13-1 (a) Box plots of hardwood concentration data. (b) Display of the model in Equation 13-1 for the completely randomized single-factor experiment.
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Graphical interpretation of the data is always useful. Box plots show the variability of the observations within a treatment (factor level) and the variability between treatments. We now discuss how the data from a single-factor randomized experiment can be analyzed statistically.
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The Analysis of Variance
Suppose we have a different 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 a treatments 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 treatment 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 Yij e i j 1, 2, p , a 1, 2, p , n (13-1)
where Yij is a random variable denoting the (ij)th observation, is a parameter common to all treatments called the overall mean, i is a parameter associated with the ith treatment called the ith treatment effect, and ij is a random error component. Notice that the model could have been written as Yij e i j 1, 2, p , a 1, 2, p , n
where i i is the mean of the ith treatment. In this form of the model, we see that each treatment de nes a population that has mean i, consisting of the overall mean plus an effect i that 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 i and 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.
Table 13-2 Typical Data for a Single-Factor Experiment Treatment 1 2 o a y11 y21 o ya1 Observations y12 y22 o ya2 p p ooo p y1n y2n yan Totals y1. y2. o ya. y.. Averages y1. y2. o ya. y. .