Table 14-32 A 27 4 Fractional Factorial Design III A B C D AB E AC F BC G ABC

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

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7 This 2III 4 design is called a saturated fractional factorial, because all the available degrees of freedom are used to estimate main effects. It is possible to combine sequences of these resolution III fractional factorials to separate the main effects from the two-factor interactions. The procedure is illustrated in Montgomery (2001) and in Box, Hunter, and Hunter (1978).

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EXERCISES FOR SECTION 14-9

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14-31. R. D. Snee ( Experimenting with a Large Number of Variables, in Experiments in Industry: Design, Analysis and Interpretation of Results, by R. D. Snee, L. D. Hare, and J. B. Trout, eds., ASQC, 1985) describes an experiment in which a 25 1 design with I ABCDE was used to investigate the effects of ve factors on the color of a chemical product. The factors are A solvent/reactant, B catalyst/reactant, C temperature, D reactant purity, and E reactant pH. The results obtained are as follows: e a b abe c ace bce abc 0.63 2.51 2.68 1.66 2.06 1.22 2.09 1.93 d ade bde abd cde acd bcd abcde 6.79 6.47 3.45 5.68 5.22 4.38 4.30 4.05 (a) Write down the alias relationships. (b) Estimate the factor effects. Which factor effects appear large (c) Project this design into a full factorial in the three apparently important factors and provide a practical interpretation of the results. 14-33. An article in Industrial and Engineering Chemistry ( More on Planning Experiments to Increase Research Ef ciency, 1970, pp. 60 65) uses a 25 2 design to investigate the effect on process yield of A condensation temperature, B amount of material 1, C solvent volume, D condensation time, and E amount of material 2. The results obtained are as follows: ae ab ad bc 23.2 15.5 16.9 16.2 cd ace bde abcde 23.8 23.4 16.8 18.1

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(a) Prepare a normal probability plot of the effects. Which factors are active (b) Calculate the residuals. Construct a normal probability plot of the residuals and plot the residuals versus the tted values. Comment on the plots. (c) If any factors are negligible, collapse the 25 1 design into a full factorial in the active factors. Comment on the resulting design, and interpret the results. 14-32. Montgomery (2001) describes a 24 1 fractional factorial design used to study four factors in a chemical process. The factors are A temperature, B pressure, C concentration, and D stirring rate, and the response is ltration rate. The design and the data are as follows:

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(a) Verify that the design generators used were I ACE and I BDE. (b) Write down the complete de ning relation and the aliases from the design. (c) Estimate the main effects. (d) Prepare an analysis of variance table. Verify that the AB and AD interactions are available to use as error. (e) Plot the residuals versus the tted values. Also construct a normal probability plot of the residuals. Comment on the results.

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Run 1 2 3 4 5 6 7 8

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Filtration Rate 45 100 45 65 75 60 80 96

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CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

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14-34. Consider the 26 2 design in Table 14-28. Suppose that after analyzing the original data, we nd that factors C and E can be dropped. What type of 2k design is left in the remaining variables 14-35. Consider the 26 2 design in Table 14-28. Suppose that after the original data analysis, we nd that factors D and F can be dropped. What type of 2k design is left in the remaining variables Compare the results with Exercise 14-34. Can you explain why the answers are different 14-36. Suppose that in Exercise 14-22 it was possible to run only a 1 fraction of the 24 design. Construct the design and 2 use only the data from the eight runs you have generated to perform the analysis. 14-37. Suppose that in Exercise 14-16 only a 1 4 fraction of the 25 design could be run. Construct the design and analyze the data that are obtained by selecting only the response for the eight runs in your design. 8 14-38. Construct the 2IV 4 design recommended in Table 13-29. What are the aliases of the main effects and two-factor interactions 14-39. Construct a 26 3 fractional factorial design. Write III down the aliases, assuming that only main effects and twofactor interactions are of interest. 14-40. Consider the problem in Exercise 14-19. Suppose that only half of the 32 runs could be made. (a) Choose the half that you think should be run. (b) Write out the alias relationships for your design. (c) Estimate the factor effects. (d) Plot the effect estimates on normal probability paper and interpret the results. (e) Set up an analysis of variance for the factors identi ed as potentially interesting from the normal probability plot in part (d). (f) Analyze the residuals from the model. (g) Provide a practical interpretation of the results.

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investigate the effect of pH and catalyst concentration on product viscosity (cSt). The data are as follows: Catalyst Concentration 2.5 2.7 pH 5.6 5.9 192, 199, 189, 198 185, 193, 185, 192 178, 186, 179, 188 197, 196, 204, 204

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(a) Test for main effects and interactions using 0.05. What are your conclusions (b) Graph the interaction and discuss the information provided by this plot. (c) Analyze the residuals from this experiment. 14-42. Heat treating of metal parts is a widely used manufacturing process. An article in the Journal of Metals (Vol. 41, 1989) describes an experiment to investigate atness distortion from heat treating for three types of gears and two heattreating times. The data are as follows: Time (minutes) Gear Type 20-tooth 24-tooth 28-tooth 90 0.0265 0.0340 0.0430 0.0510 0.0405 0.0575 120 0.0560 0.0650 0.0720 0.0880 0.0620 0.0825

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

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