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www.downloadslide.com 12.4 Some Elementary Experimental Designs 651 12.17 Refer to the matched-pairs experiment and assume that the ith measurement (i = 1, 2),inthe jth pair, where j = 1, 2,...,n,is Y ij = µ i + U j + ε ij , where µ i = expected response for population i, where i = 1, 2, U j = a random variable that is uniformly distributed on the interval (−1, +1), ε ij = random error associated **with** the ith measurement in the jth pair. Assume that the ε ij ’s are independent normal random variables **with** E(ε ij ) = 0 and V (ε ij ) = σ 2 , and that U j and ε ij are independent. a Find E(Y ij ). b Argue that the Y 1 j ’s, for j = 1, 2,...,n, are not normally distributed. (There is no need to actually find the distribution of the Y 1 -values.) c Show that Cov(Y 1 j , Y 2 j ) = 1/3, for j = 1, 2,...,n. d Show that D j = Y 1 j − Y 2 j are independent, normally distributed random variables. e In parts (a)–(d), you verified that the differences **with**in each pair can be normally distributed even though the individual measurements **with**in the pairs are not. Can you come up **with** another example that illustrates this same phenomenon? 12.4 Some Elementary Experimental Designs In Chapters 8 and 10, we considered methods to compare the means of two populations based on independent random samples obtained from each. Section 12.3 dealt **with** a comparison of two population means through the matched-pairs experiment. In this section, we present general considerations associated **with** designing experiments. Specifically, we consider extensions of the independent samples and matched-pairs methodologies when the objective is to compare the means of more than two populations. Suppose that we wish to compare five teaching techniques, A, B, C, D, and E, and that we use 125 students in the study. The objective is to compare the mean scores on a standardized test for students taught by each of the five methods. How would we proceed? Even though the 125 students are in some sense representative of the students that these teaching methods target, are the students all identical? The answer is obviously no. There are likely to be boys and girls in the group, and the methods might not be equally effective for both genders. There are likely to be differences in the native abilities of the students in the group, resulting in some students performing better regardless of the teaching method used. Different students may come from families that place different emphases on education, and this could have an impact on the scores on the standardized test. In addition, there may be other differences among the 125 students that would have an unanticipated effect on the test scores. Based on these considerations, we decide that it might be wise to randomly assign 25 students to each of five groups. Each group will be taught using one of the techniques under study. The random division of the students into the five groups achieves two objectives. First, we eliminate the possible biasing effect of individual

www.downloadslide.com 652 Chapter 12 Considerations in Designing Experiments characteristics of the students on the measurements that we make. Second, it provides a probabilistic basis for the selection of the sample that permits the statistician to calculate probabilities associated **with** the observations in the sample and to use these probabilities in making inferences. The preceding experiment illustrates the basic components of a designed experiment. The experimental units in this study are the individual students. DEFINITION 12.1 Experimental units are the objects upon which measurements are taken. This experiment involves a single factor—namely, method of teaching. In this experiment, the factor has five levels: A,B,C,D,andE. DEFINITION 12.2 Factors are variables completely controlled by the experimenter. The intensity level (distinct subcategory) of a factor is called its level. In a single-factor experiment like the preceding one, each level of the single factor represents a treatment. Thus, in our education example, there are five treatments, one corresponding to each of the teaching methods. As another example, consider an experiment conducted to investigate the effect of various amounts of nitrogen and phosphate on the yield of a variety of corn. An experimental unit would be a specified acreage—say, 1 acre—of corn. A treatment would be a fixed number of pounds of nitrogen x 1 and of phosphate x 2 applied to a given acre of corn. For example, one treatment might be to use x 1 = 100 pounds of nitrogen per acre and x 2 = 200 pounds of phosphate. A second treatment might correspond to x 1 = 150 and x 2 = 100. Notice that the experimenter could use different amounts (x 1 , x 2 ) of nitrogen and phosphate and that each combination would represent a different treatment. DEFINITION 12.3 A treatment is a specific combination of factor levels. The preceding experiment for comparing teaching methods A, B, C, D, and E entailed randomly dividing the 125 students into five groups, each of size 25. Each group received exactly one of the treatments. This is an example of a completely randomized design. DEFINITION 12.4 A completely randomized design to compare k treatments is one in which a group of n relatively homogeneous experimental units are randomly divided into k subgroups of sizes n 1 , n 2 ,...,n k (where n 1 + n 2 + ··· + n k = n). All experimental units in each subgroup receive the same treatment, **with** each treatment applied to exactly one subgroup.

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