Question

1. An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows: Treatment Replicate A B CCombinationI 22 31 32 43 35 34 55 47 44 45 40 37 60 50 54 39 41 25 29 50 46 38 36 ac bc abc 47 (a) Estimate the factor effects. Which effects appear to be large? (b) Use the analysis of variance to confirm your conclusions for part (a) (c) Write down a regression model for predicting tool life (in hours) based on the results of this experiment. (d) Use the regression model in (c) 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? (e) Analyze the residuals. Are there any obvious problems? (f) Based on the analysis of main effects and interaction plots, what levels of A, B, and C would you recommend using?

Answer 1

Date:--12/05/2019 Answer:- (a) Estimate the factorial effects. Which effects appear to be large (significant)? The estimates of the factorial effects are summarized in the table below Effect Estimate 0.3333 11.3333 6.8333 AB-1.6667 AC -8.8333 BC-2.833;3 ABC -2.1667 The effects B, C and AC appear to be large (or significant), based on the table above (b) Use analysis of variance to confirm your conclusions for part (a) The results of analysis of variance in SAS are given below. Note that the effects B, C, and AC are all significant Analysis of Variance ean SumO Squares Square F Value Pr > F Source 7 1612.66667 482.66667 2095. 33333 Model ErroI Corrected Total 230.38095 30. 16667 7.64 0.0004 16 23 0.7696 0.6689 5.49242 R 40.83333 13.45082 Root MSE Dependent Mean Coeff Var R-Square Adj R-Sq

Parameter Estimate Standard Errort Value Pr > ltl Variable DF K.0001 0.8837 0.0001 0.0077 0.4681 0.0012 0.2245 0.3483 40.83333 0.16667 5.66667 3.41667 -0.83333 -4. 41667 1.41667 -1.08333 1.12114 1.12114 1.12114 1.12114 1.12114 1.12114 1.12114 1.12114 36.42 0.15 5.05 3.05 -0.74 -3.94 Intercept x1x2 2X3 x1x2x3 -0.97 (c) Write down a regression model for predicting tool life (in hours) based on the results of this experiment Based on the results above, the suggested model for predicting tool life (in hours) is: where. Yijk1 is the tool life. A: ±1. Bj = ±1. Ck = ±1. 1 = 1, 2, 3, and Eijk 1 ~ N(0,1) are i.i.d random variables Using just the significant effects, A, C, and AC, together with the main effect of A, the estimated regression function is given as Y- 40.833330.16667A 5.66667B 3.41667C-4.41667 AC where Y is the etch rate and A = ±1, B = ±1, and C = ±1. since Cis already included in the model, whet her A should be included or not is usually subjective

(d) Use the regression model to generate cont our plots of the tool life response. Do they provide insight regarding the desirable operating conditions for this process? The following figures are the contour plots of the tool life response. On the left is the contour plot where B = z2 1 and on the right is the contour plot with B = z2 =-1. Note that in both cases the contour plots suggest that the factor levels should be set so that A = zı =-1 and C = z3-1, in order to maximize the tool life

江-: For both levels of B, the optimal setting of the factor levels is the same. That is, (A, C) ()

(e) Analyze the residuals. Are there any obvious problems? Several plots of the residuals were created. First, a plot of the residuals versus the predicted values was generated. This plot is given here: When the residuals are plotted against the predicted values, the evidence against the assumptions of normality and constancy of error variance is weak. The following three plots are the plots of the residuals against factors A, B and C, respectively.

An examination of these plots suggests that the assumption of normally distributed errors is reasonable. There is one value with a particularly large residual errorse rly large residual.ption of

A histogr am and a QQ-plot of the residuals from the model in part (c) are given here These plots show the distribution of the residuals may not be symmetric. This is particularly evident in the histogram. The QQ-plot shows some possible deviation from normality. However, neither of these plots gives substantial evidence that the data are not normally distributed. This was checked by conducting formal tests for nonnorm ality. The results from SAS are given below Goodness-of-Fit Tests for Normal Distribution Test Kolmogorov-Smirnov D 0.13290973 Pr> D 0.150 Anderson-Darling n Mises Sq 0.05982132 Pr > W-Sq 0.250 A-Sq 0.40198755 Pr > A-Sq >0.250

These tests do not show a significant deviation from normality. We accept the hypothesis that the ata are normally distributed (f) Generate main effect and interaction plots for significant effects. Based on them, what levels of A, B, and Cwould vou recommend using The following plots are the main effect plots for A and C I'

The interaction plot for A and C is: Based on these plots, we see that the mean response is highest if (A, B,C)(-+,+). Since the response is the tool life, long tool life is optimal. This is the setting suggested by these plots