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Solution HW3

# Solution HW3 - Designed Experimentation Solutions for HW3...

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Designed Experimentation Solutions for HW3 6.7 An experiment was performed to improve the yield of a chemical process. Four factor were selected, and two replicates of a completely randomized experiment were run. The results are shown in the following table: a) Estimate the factor effects. Estimated Effects and Coefficients for C9 (coded units) Term Effect Coef SE Coef T P Constant 82.781 0.4891 169.24 0.000 A -9.063 -4.531 0.4891 -9.26 0.000 B -1.313 -0.656 0.4891 -1.34 0.198 C -2.687 -1.344 0.4891 -2.75 0.014 D 3.937 1.969 0.4891 4.02 0.001 A*B 4.062 2.031 0.4891 4.15 0.001 A*C 0.688 0.344 0.4891 0.70 0.492 A*D -2.187 -1.094 0.4891 -2.24 0.040 B*C -0.563 -0.281 0.4891 -0.57 0.573 B*D -0.188 -0.094 0.4891 -0.19 0.850 C*D 1.688 0.844 0.4891 1.72 0.104 A*B*C -5.187 -2.594 0.4891 -5.30 0.000 A*B*D 4.687 2.344 0.4891 4.79 0.000 A*C*D -0.938 -0.469 0.4891 -0.96 0.352 B*C*D -0.938 -0.469 0.4891 -0.96 0.352 A*B*C*D 2.437 1.219 0.4891 2.49 0.024 S = 2.76699 R-Sq = 92.47% R-Sq(adj) = 85.42% Answer: There are a number of significant terms (w/ P values LE 0.05): A, C, D, AB, AD, ABC, ABD, ABCD. Since factor B is included in an interaction, it will also be added to the model in the next step. b) Prepare an analysis of variance table, and determine which factors are important in explaining yield. Term Effect Coef SE Coef T P Constant 82.781 0.4855 170.51 0.000 A -9.063 -4.531 0.4855 -9.33 0.000 B -1.312 -0.656 0.4855 -1.35 0.190 C -2.687 -1.344 0.4855 -2.77 0.011 D 3.937 1.969 0.4855 4.06 0.001 A*B 4.062 2.031 0.4855 4.18 0.000 A*D -2.187 -1.094 0.4855 -2.25 0.035 A*B*C -5.188 -2.594 0.4855 -5.34 0.000 A*B*D 4.688 2.344 0.4855 4.83 0.000 A*B*C*D 2.437 1.219 0.4855 2.51 0.020 S = 2.74638 R-Sq = 89.80% R-Sq(adj) = 85.63%

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Analysis of Variance for C9 (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 4 852.63 852.63 213.156 28.26 0.000 2-Way Interactions 2 170.31 170.31 85.156 11.29 0.000 3-Way Interactions 2 391.06 391.06 195.531 25.92 0.000 4-Way Interactions 1 47.53 47.53 47.531 6.30 0.020 Residual Error 22 165.94 165.94 7.543 Lack of Fit 6 43.44 43.44 7.240 0.95 0.491 Pure Error 16 122.50 122.50 7.656 Total 31 1627.47 Unusual Observations for C9 Obs StdOrder C9 Fit SE Fit Residual St Resid 13 13 99.0000 92.6875 1.5353 6.3125 2.77R Answer: The same factors and interactions are still influential as were found in part a. Factor B does not contribute to the overall model, (i.e., is not statistically significant at the a=0.05 level) but has been added to maintain the hierarchy of effects. c) Write down a regression model for predicting yield, assuming that all four factors were varied over the range from -1 to +1 (in coded units). ABCD ABD ABC AD AB D C A X X X X X X X X y 22 . 1 34 . 2 59 . 2 09 . 1 03 . 2 97 . 1 34 . 1 53 . 4 78 . 82 ˆ + + - - + + - - = Answer: The regression model appears above. Coefficients were obtained from the Minitab analysis above. d) Plot the residuals vs. predicted yield and a normal probability plot. Does the residual analysis appear satisfactory?
Standardized Residual Percent 3 2 1 0 -1 -2 -3 99 95 90 80 70 60 50 40 30 20 10 5 1 Normal Probability Plot of the Residuals (response is C9) Fitted Value Standardized Residual 100 95 90 85 80 75 70 3 2 1 0 -1 -2 Residuals Versus the Fitted Values (response is C9) Answer: Both the NPP and Residuals vs. Fitted values appear to support the ANOVA analysis assumptions of e~NID(0, σ 2 ). The NPP passes the fat pen test indicating

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normality and the scatter plot has a shotgun pattern indicating independently distributed residuals with mean equal to zero and constant variance. e) Two three-factor interactions, ABC and ABD, apparently have large effects. Draw a cube plot in the factors A, B and C with the average yields shown at each corner. Repeat using the factors A, B and D. Do these two plots aid in data interpretation? Where would you recommend that the process be run with respect to the four variables?
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