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www.downloadslide.com Exercises 685 a Is there evidence of a difference in mean wear among the four treatments? Use α = .05. b Estimate the mean difference in haze increase between treatments B and C, using a 99% confidence interval. c Find a 90% confidence interval for the mean wear for lenses receiving treatment A. 13.31 With the ongoing energy crisis, researchers for the major oil companies are attempting to find alternative sources of oil. It is known that some types of shale contain small amounts of oil that feasibly (if not economically) could be extracted. Four methods have been developed for extracting oil from shale, and the government has decided that some experimentation should be done to determine whether the methods differ significantly in the average amount of oil that each can extract from the shale. Method 4 is known to be the most expensive method to implement, and method 1 is the least expensive, so inferences about the differences in performance of these two methods are of particular interest. Sixteen bits of shale (of the same size) were randomly subjected to the four methods, **with** the results shown in the accompanying table (the units are in liters per cubic meter). All inferences are to be made **with** α = .05. Method 1 Method 2 Method 3 Method 4 3 2 5 5 2 2 2 2 1 4 5 4 2 4 1 5 a b Assuming that the 16 experimental units were as alike as possible, implement the appropriate ANOVA to determine whether there is any significant difference among the mean amounts extracted by the four methods. Use α = .05. Set up a 95% confidence interval for the difference in the mean amounts extracted by the two methods of particular interest. Interpret the result. 13.32 Refer to Exercise 13.14. Construct a 95% confidence interval for the mean amount of residue from DDT. 13.33 Refer to Exercise 13.15. Compare the mean dissolved oxygen content in midstream above the plant **with** the mean content adjacent to the plant (location 2 versus location 3). Use a 95% confidence interval. 13.34 Refer to Exercise 13.15. Compare the mean dissolved oxygen content for the two locations above the plant **with** the mean content slightly downriver from the plant, by finding a 95% confidence interval for (1/2)(µ 1 + µ 2 ) − µ 4 . 13.35 Refer to Exercise 13.16. The average increase in heart rate for the ten individuals in each age category were Average Heart Age Sample Size Rate Increase 10–19 10 30.9 20–39 10 27.5 40–59 10 29.5 60–69 10 28.2 a b Find a 90% confidence interval for the difference in mean increase in heart rate for the 10–19 and 60–69 age groups. Find a 90% confidence interval for the mean increase in heart rate for the 20–39 age group.

686 Chapter 13 The Analysis of Variance www.downloadslide.com 13.8 A Statistical Model for the Randomized Block Design The method for constructing a randomized block design was presented in Section 12.4. As previously indicated in Definition 12.6, the randomized block design is a design for comparing k treatments using b blocks. The blocks are selected so that, hopefully, the experimental units **with**in each block are essentially homogeneous. The treatments are randomly assigned to the experimental units in each block in such a way that each treatment appears exactly once in each of the b blocks. Thus, the total number of observations obtained in a randomized block design is n = bk. Implicit in the consideration of a randomized block design is the presence of two qualitative independent variables, “blocks” and “treatments.” In this section, we present a formal statistical model for the randomized block design. Statistical Model for a Randomized Block Design For i = 1, 2,...,k and j = 1, 2,...,b, Y ij = µ + τ i + β j + ε ij where Y ij = the observation on treatment i in block j, µ = the overall mean, τ i = the nonrandom effect of treatment i, where ∑ k i=1 τ i = 0, β j = the nonrandom effect of block j, where ∑ b j=1 β j = 0. ε ij = random error terms such that ε ij are independent normally distributed random variables **with** E(ε ij ) = 0 and V (ε ij ) = σ 2 . Notice that µ, τ 1 ,τ 2 ,...,τ k , and β 1 ,β 2 ,...,β b are all assumed to be unknown constants. This model differs from that for the completely randomized design (a specific type of one-way layout) only in containing parameters associated **with** the different blocks. Because the block effects are assumed to be fixed but unknown, this model usually is referred to as the fixed block effects model. A random block effects model, another model for the randomized block design in which the β’s are assumed to be random variables, is considered in the supplementary exercises. Our formal development in the body of this text is restricted to the fixed block effects model. The statistical model just presented very clearly summarizes all the assumptions made in the analysis of data in a randomized block design **with** fixed block effects. Let us consider the observation Y ij made on treatment i in block j. Notice that the assumptions in the model imply that E(Y ij ) = µ + τ i + β j and V (Y ij ) = σ 2 for i = 1, 2,...,k and j = 1, 2,...,b. Let us consider the observations made on treatment i and observe that two observations receiving treatment i have means that

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