CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

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MIND-EXPANDING EXERCISES

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14-55. Consider an unreplicated 2k factorial, and suppose that one of the treatment combinations is missing. One logical approach to this problem is to estimate the missing value with a number that makes the highest order interaction estimate zero. Apply this technique to the data in Example 14-5, assuming that ab is missing. Compare the results of the analysis of these data with the results in Example 14-5. 14-56. What blocking scheme would you recommend if it were necessary to run a 24 design in four blocks of four runs each 14-57. Consider a 22 design in two blocks with AB confounded with blocks. Prove algebraically that SSAB SSBlocks. 14-58. Consider a 23 design. Suppose that the largest number of runs that can be made in one block is four, but we can afford to perform a total of 32 observations. (a) Suggest a blocking scheme that will provide some information on all interactions. (b) Show an outline (source of variability, degrees of freedom only) for the analysis of variance for this design. 14-59. Construct a 25 1 design. Suppose that it is necessary to run this design in two blocks of eight runs each. Show how this can be done by confounding a twofactor interaction (and its aliased three-factor interaction) with blocks. 14-60. Construct a 27 2 design. Show how this IV design may be confounded in four blocks of eight runs each. Are any two-factor interactions confounded with blocks 14-61. Construct a 27 3 design. Show how this deIV sign can be confounded in two blocks of eight runs each without losing information on any of the two-factor interactions. 14-62. Set up a 27 4 design using D AB, E AC, III F BC, and G ABC as the design generators. Ignore all interaction above the two factors. (a) Verify that each main effect is aliased with three two-factor interactions. (b) Suppose that a second 27 4 design with generators III D AB, E AC, F BC, and G ABC is run. What are the aliases of the main effects in this design (c) What factors may be estimated if the two sets of factor effect estimates above are combined To work Exercises 14-63 through 14-67 you will need to read Section 14.6 on the CD. 14-63. Consider the experiment described in Example 14-4. Suppose that both factors were random. (a) Analyze the data and draw appropriate conclusions. (b) Estimate the variance components. 14-64. For the breaking strength data in Table S14-1, suppose that the operators were chosen at random, but machines were a xed factor. Does this in uence the analysis or your conclusions 14-65. A company employs two time-study engineers. Their supervisor wishes to determine whether the standards set by them are in uenced by an interaction between engineers and operators. She selects three operators at random and conducts an experiment in which the engineers set standard times for the same job. She obtains the data shown here:

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Operator Engineer 1 2 1 2.59 2.78 2.15 2.86 2 2.38 2.49 2.85 2.72 3 2.40 2.72 2.66 2.87

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(a) State the appropriate hypotheses. (b) Use the analysis of variance to test these hypotheses with 0.05. (c) Graphically analyze the residuals from this experiment. (d) Estimate the appropriate variance components. 14-66. Consider the experiment on baked anode density described in Exercise 14-4. Suppose that positions on the furnace were chosen at random and temperature is a xed factor. (a) State the appropriate hypotheses. (b) Use the analysis of variance to test these hypotheses with 0.05. (c) Estimate the variance components. 14-67. Consider the experiment described in Exercise 14-63. How does the analysis (and conclusions) change if both factors are random Use 0.05. To work Exercises 14-68 and 14-69 you will need to read Section 14-7.4 on the CD.

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14-10 RESPONSE SURFACE METHODS AND DESIGNS (CD ONLY)

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