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Ch06 Design Of Experiments (Texas Tech University) Scan to open on Studocu Studocu is not sponsored or endorsed by any college or university Ch06 Design Of Experiments (Texas Tech University) Scan to open on Studocu Studocu is not sponsored or endorsed by any college or university Downloaded by Omar Hassan (ayssar.omar.borham@hotmail.com) lOMoARcPSD|39623336

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments , Wiley, NY 6-1 Chapter 6 The 2 k Factorial Design Solutions 6.1. In a 2 4 factorial design, the number of degrees of freedom for the model, assuming the complete factorial model, is (a) 7 (b) 5 (c) 6 (d) 11 (e) 12 (f) None of the above 6.2. A 2 3 factorial is replicated twice. The number of pure error or residual degrees of freedom are (a) 4 (b) 12 (c) 15 (d) 2 (e) 8 (f) None of the above 6.3. A 2 3 factorial is replicated twice. The ANOVA indicates that all main effects are significant but the interactions are not significant. The interaction terms are dropped from the model. The number of residual degrees of freedom for the reduced model are (a) 12 (b) 8 (c) 16 (d) 14 (e) 10 (f) None of the above 6.4. A 2 3 factorial is replicated three times. The ANOVA indicates that all main effects are significant, but two of the interactions are not significant. The interaction terms are dropped from the model. The number of residual degrees of freedom for the reduced model are (a) 12 (b) 14 (c) 6 (d) 10 (e) 8 (f) None of the above Downloaded by Omar Hassan (ayssar.omar.borham@hotmail.com) lOMoARcPSD|39623336

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments , Wiley, NY 6-2 6.5. An engineer is interested in the effects of cutting speed ( A ), tool geometry ( B ), and cutting angle on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 2 3 factorial design are run. The results are as follows: Treatment Replicate A B C Combination I II III - - - (1) 22 31 25 + - - a 32 43 29 - + - b 35 34 50 + + - ab 55 47 46 - - + c 44 45 38 + - + ac 40 37 36 - + + bc 60 50 54 + + + abc 39 41 47 (a) Estimate the factor effects. Which effects appear to be large? From the normal probability plot of effects below, factors B , C , and the AC interaction appear to be significant. (b) Use the analysis of variance to confirm your conclusions for part (a). The analysis of variance confirms the significance of factors B , C , and the AC interaction. Design Expert Output Response: Life in hours ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 1612.67 7 230.38 7.64 0.0004 significant A 0.67 1 0.67 0.022 0.8837 B 770.67 1 770.67 25.55 0.0001 C 280.17 1 280.17 9.29 0.0077 AB 16.67 1 16.67 0.55 0.4681 Downloaded by Omar Hassan (ayssar.omar.borham@hotmail.com) lOMoARcPSD|39623336

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments , Wiley, NY 6-3 AC 468.17 1 468.17 15.52 0.0012 BC 48.17 1 48.17 1.60 0.2245 ABC 28.17 1 28.17 0.93 0.3483 Pure Error 482.67 16 30.17 Cor Total 2095.33 23 The Model F-value of 7.64 implies the model is significant. There is only a 0.04% chance that a "Model F-Value" this large could occur due to noise. The reduced model ANOVA is shown below. Factor A was included to maintain hierarchy. Design Expert Output Response: Life in hours ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 1519.67 4 379.92 12.54 < 0.0001 significant A 0.67 1 0.67 0.022 0.8836 B 770.67 1 770.67 25.44 < 0.0001 C 280.17 1 280.17 9.25 0.0067 AC 468.17 1 468.17 15.45 0.0009 Residual 575.67 19 30.30 Lack of Fit 93.00 3 31.00 1.03 0.4067 not significant Pure Error 482.67 16 30.17 Cor Total 2095.33 23 The Model F-value of 12.54 implies the model is significant. There is only a 0.01% chance that a "Model F-Value" this large could occur due to noise. Effects B, C and AC are significant at 1%. (c) Write down a regression model for predicting tool life (in hours) based on the results of this experiment. 40.8333 0.1667 5.6667 3.4167 4.4167 ijk A B C A C y x x x x x = + + + + Design Expert Output Coefficient Standard 95% CI 95% CI Factor Estimate DF Error Low High VIF Intercept 40.83 1 1.12 38.48 43.19 A-Cutting Speed 0.17 1 1.12 -2.19 2.52 1.00 B-Tool Geometry 5.67 1 1.12 3.31 8.02 1.00 C-Cutting Angle 3.42 1 1.12 1.06 5.77 1.00 AC -4.42 1 1.12 -6.77 -2.06 1.00 Final Equation in Terms of Coded Factors: Life = +40.83 +0.17 * A +5.67 * B +3.42 * C -4.42 * A * C Final Equation in Terms of Actual Factors: Life = +40.83333 +0.16667 * Cutting Speed +5.66667 * Tool Geometry +3.41667 * Cutting Angle -4.41667 * Cutting Speed * Cutting Angle Downloaded by Omar Hassan (ayssar.omar.borham@hotmail.com) lOMoARcPSD|39623336

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments , Wiley, NY 6-4 The equation in part (c) and in the given in the computer output form a “hierarchical” model, that is, if an interaction is included in the model, then all of the main effects referenced in the interaction are also included in the model. (d) Analyze the residuals. Are there any obvious problems? There is nothing unusual about the residual plots. (e) Based on the analysis of main effects and interaction plots, what levels of A , B , and C would you recommend using? Since B has a positive effect, set B at the high level to increase life. The AC interaction plot reveals that life would be maximized with C at the high level and A at the low level. 6.6. Reconsider part (c) of Problem 6.5. Use the regression model to generate response surface and contour plots of the tool life response. Interpret these plots. Do they provide insight regarding the desirable operating conditions for this process? Downloaded by Omar Hassan (ayssar.omar.borham@hotmail.com) lOMoARcPSD|39623336

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments , Wiley, NY 6-5 The response surface plot and the contour plot in terms of factors A and C with B at the high level are shown below. They show the curvature due to the AC interaction. These plots make it easy to see the region of greatest tool life. 6.7. Find the standard error of the factor effects and approximate 95 percent confidence limits for the factor effects in Problem 6.5. Do the results of this analysis agree with the conclusions from the analysis of variance? ( ) 2 ( ) 2 3 2 1 1 30.17 2.24 2 3 2 effect k SE S n - - = = = Variable Effect A 0.333 B 11.333 * AB -1.667 C 6.833 * AC -8.833 * BC -2.833 ABC -2.167 The 95% confidence intervals for factors B, C and AC do not contain zero. This agrees with the analysis of variance approach. 6.8. Plot the factor effects from Problem 6.5 on a graph relative to an appropriately scaled t distribution. Does this graphical display adequately identify the important factors? Compare the conclusions from this plot with the results from the analysis of variance. 30.17 3.17 3 E MS S n = = = Downloaded by Omar Hassan (ayssar.omar.borham@hotmail.com) lOMoARcPSD|39623336

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