hw1010f11solu

# hw1010f11solu - Stat 514 Homework 10 1. An engineer is...

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Stat 514 Homework 10 1. An engineer is interested in the eﬀects of cutting speed( A ), tool geometry ( B ), and cutting angle ( C )onthe life (in hours) of a machine tool. Two levels of each factor are chose, and three replicates of a 2 3 factorial design are run. The results follow. Factor Replicate ABC II I I --- 22 31 25 + -- 32 43 29 - + - 35 34 50 ++ - 55 47 46 +4 44 53 8 + - 03 73 6 - + + 60 50 54 +++3 94 14 7 (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 A 0 . 3333 B 11 . 3333 C 6 . 8333 AB - 1 . 6667 AC - 8 . 8333 BC - 2 . 8333 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 Sum of Mean Source DF Squares Square F Value Pr > F Model 7 1612.66667 230.38095 7.64 0.0004 Error 16 482.66667 30.16667 Corrected Total 23 2095.33333 Root MSE 5.49242 R-Square 0.7696 Dependent Mean 40.83333 Adj R-Sq 0.6689 Coeff Var 13.45082 Parameter Estimates 1

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Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 40.83333 1.12114 36.42 <.0001 x1 1 0.16667 1.12114 0.15 0.8837 x2 1 5.66667 1.12114 5.05 0.0001 x3 1 3.41667 1.12114 3.05 0.0077 x1x2 1 -0.83333 1.12114 -0.74 0.4681 x1x3 1 -4.41667 1.12114 -3.94 0.0012 x2x3 1 -1.41667 1.12114 -1.26 0.2245 x1x2x3 1 -1.08333 1.12114 -0.97 0.3483 (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: Y ijk‘ = β 0 + β 1 A i + β 2 B j + β 3 C k + β 13 A i C k + ± ijk‘ , where, Y ijk‘ is the tool life, A i = ± 1, B j = ± 1, C k = ± 1, =1 , 2 , 3, and ± ijk‘ 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: ˆ Y =40 . 83333 + 0 . 16667 A +5 . 66667 B +3 . 41667 C - 4 . 41667 AC, where Y is the etch rate and A = ± 1, B = ± 1, and C = ± 1. since C is already included in the model, whether A should be included or not is usually subjective. (d) 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.
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## This note was uploaded on 12/09/2011 for the course STAT 514 taught by Professor Staff during the Fall '08 term at Purdue University.

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hw1010f11solu - Stat 514 Homework 10 1. An engineer is...

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