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Assignment_Solution_Chp_6

Assignment_Solution_Chp_6 - Chapter 6 Solutions 6.1 An...

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1 Chapter 6– Solutions 6.1. 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 AC 468.17 1 468.17 15.52 0.0012
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2 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. C A C B A ijk x x . x . x . x . . y 4167 4 4167 3 6667 5 1667 0 8333 40 + + + + = 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 The equation in part (c) and in the given in the computer output form a “hierarchial” 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.
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3 (d) Analyze the residuals. Are there any obvious problems? Residual Norm al % probability Normal plot of residuals -7.33333 -2.625 2.08333 6.79167 11.5 1 5 10 20 30 50 70 80 90 95 99 Predicted Residuals Residuals vs. Predicted -7.33333 -2.625 2.08333 6.79167 11.5 27.17 33.92 40.67 47.42 54.17 There is nothing unusual about the residual plots.
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