实验设计分析第六版美 道格拉斯蒙哥马利课后作业4.docx

1、实验设计分析第六版美 道格拉斯蒙哥马利课后作业4实验设计分析作业4 机械电子工程 Home Work #4 (Class note 105, 106)1. Montgomery Problem 6-182. Montgomery Problem 6-26 3. Montgomery Problem 8-4英文版题目6-18（a）Factorial Fit: Etch rate versus A极距, B压强, C气流, D功率 Estimated Effects and Coefficients for Etch rate (coded units)Term Effect CoefConsta

2、nt 776.06A极距 -101.62 -50.81B压强 -1.62 -0.81C气流 7.37 3.69D功率 306.13 153.06A极距*B压强 -7.88 -3.94A极距*C气流 -24.88 -12.44A极距*D功率 -153.63 -76.81B压强*C气流 -43.87 -21.94B压强*D功率 -0.63 -0.31C气流*D功率 -2.13 -1.06A极距*B压强*C气流 -15.63 -7.81A极距*B压强*D功率 4.12 2.06A极距*C气流*D功率 5.62 2.81B压强*C气流*D功率 -25.38 -12.69A极距*B压强*C气流*D功率

3、-40.13 -20.06Effects Plot for Etch rate According to Figure , Alpha=0.05 ,the factors of A, AD and D seem significant.Main Effects Plot (data means) for Etch rate Interaction Plot (data means) for Etch rate 结论：从主因子效应图看出A,D因子的效应比较显著；从因子交互作用看出，A*D的交互作用比其他的任何项的交互效应都大得多。（b）General Linear Model: Etch rat

4、e versus A极距, D功率 Factor Type Levels ValuesA极距 fixed 2 -1, 1D功率 fixed 2 -1, 1Analysis of Variance for Etch rate, using Adjusted SS for TestsSource DF Seq SS Adj SS Adj MS F PA极距 1 41311 41311 41311 23.77 0.000D功率 1 374850 374850 374850 215.66 0.000A极距*D功率 1 94403 94403 94403 54.31 0.000Error 12 2085

5、8 20858 1738Total 15 531421S = 41.6911 R-Sq = 96.08% R-Sq(adj) = 95.09%(c)Factorial Fit: Etch rate versus A极距, D功率 Estimated Effects and Coefficients for Etch rate (coded units)Term Effect Coef SE Coef T PConstant 776.06 10.42 74.46 0.000A极距 -101.62 -50.81 10.42 -4.88 0.000D功率 306.12 153.06 10.42 14

6、.69 0.000A极距*D功率 -153.62 -76.81 10.42 -7.37 0.000S = 41.6911 R-Sq = 96.08% R-Sq(adj) = 95.09%Analysis of Variance for Etch rate (coded units)Source DF Seq SS Adj SS Adj MS F PMain Effects 2 416161 416161 208080 119.71 0.0002-Way Interactions 1 94403 94403 94403 54.31 0.000Residual Error 12 20858 208

7、58 1738 Pure Error 12 20858 20858 1738Total 15 531421So we get Regression: Y = 776.06 - 50.81A + 153.06D - 76.81A*D(d)Residual Plots for Etch rate 残差分析：从正态概率图中，残差紧密的分布在直线的两侧，呈线性分布，很好的吻合了正态性分布假定；残差关于拟合值也是上下基本大致对称的分布的，证明回归模型的方程结论是非常符合这个实验数据统计理论的。（e）根据前面的分析可知，A,D的影响效应是B,C的好几倍，B,C的影响基本可以忽略了。剩余的两个因子A,D是非

8、常重要的，变成重复22=4次.但要注意实验次序应重新随机化排列。A极距D功率Etch rate-1-15501-1669-1-16041-1650-1-16331-1642-1-16011-1635-11103711749-11105211868-11107511860-11106311729General Linear Model: Etch rate versus A极距, D功率 Factor Type Levels ValuesA极距 fixed 2 -1, 1D功率 fixed 2 -1, 1Analysis of Variance for Etch rate, using Adju

9、sted SS for TestsSource DF Seq SS Adj SS Adj MS F PA极距 1 41311 41311 41311 23.77 0.000D功率 1 374850 374850 374850 215.66 0.000A极距*D功率 1 94403 94403 94403 54.31 0.000Error 12 20858 20858 1738Total 15 531421S = 41.6911 R-Sq = 96.08% R-Sq(adj) = 95.09%（f） Interaction Plot (data means) for Etch rate 表明当A

10、低水平，D高水平时，二者的交互作用非常明显。(g)Residuals vs Order for Etch rate 我发现残差有峰有谷近似沿着某条正弦曲线分布的趋势。题目6-26（a） According to the following Figure , Alpha=0.05 ,the factors of A, A*B and C seem significant for M -W only.(b)Factorial Fit: Molecular Weight versus A, B, C, D Estimated Effects and Coefficients for Molecula

11、r Weight (coded units)Term Effect Coef SE Coef T PConstant 2506.25 11.597 216.12 0.000A 123.65 61.83 11.596 5.33 0.006B -11.28 -5.64 11.596 -0.49 0.652C 201.21 100.60 11.596 8.68 0.001D 6.17 3.09 11.596 0.27 0.803A*B 120.29 60.14 11.594 5.19 0.007A*C 20.48 10.24 11.590 0.88 0.427A*D -16.64 -8.32 11.

12、575 -0.72 0.512B*C -22.37 -11.18 11.596 -0.96 0.389B*D 7.73 3.87 11.595 0.33 0.755C*D 12.89 6.45 11.592 0.56 0.608A*B*C 14.82 7.41 11.536 0.64 0.556A*B*D -13.82 -6.91 11.398 -0.61 0.577A*C*D -23.04 -11.52 11.037 -1.04 0.356B*C*D 2.58 1.29 11.556 0.11 0.916A*B*C*D -9.63 -4.81 4.524 -1.06 0.347S = 46.

13、3860 R-Sq = 97.17% R-Sq(adj) = 86.54%Analysis of Variance for Molecular Weight (coded units)Source DF Seq SS Adj SS Adj MS F PMain Effects 4 138043 223768 55942.0 26.00 0.0042-Way Interactions 6 152348 63589 10598.2 4.93 0.0723-Way Interactions 4 2311 4084 1021.0 0.47 0.7564-Way Interactions 1 2437

14、2437 2436.8 1.13 0.347Residual Error 4 8607 8607 2151.7 Lack of Fit 1 782 782 781.7 0.30 0.622 Pure Error 3 7825 7825 2608.3Total 19 303745 结论：从上面的分析中看出A、C、A*B的P值均小于0.05，故A、C、A*B对Molecular Weight的影响最大。因为有A,B的交互作用明显，所以暗示了回归模型的方程不是线性的，是曲线形状的。(c)Factorial Fit: Molecular Weight versus A, B, C Estimated

15、Effects and Coefficients for Molecular Weight (coded units)Term Effect Coef SE Coef T PConstant 2505.57 10.781 232.40 0.000A 138.67 69.33 10.250 6.76 0.000B -7.18 -3.59 10.747 -0.33 0.743C 208.03 104.02 10.677 9.74 0.000A*B 75.25 37.62 3.860 9.75 0.000S = 43.1422 R-Sq = 90.81% R-Sq(adj) = 88.36%Anal

16、ysis of Variance for Molecular Weight (coded units)Source DF Seq SS Adj SS Adj MS F PMain Effects 3 99017 273889 91296 49.05 0.0002-Way Interactions 1 176810 176810 176810 95.00 0.000Residual Error 15 27919 27919 1861 Lack of Fit 4 13869 13869 3467 2.71 0.085 Pure Error 11 14050 14050 1277Total 19 3

17、03745Regression Equation: Y = 2505.57+69.33A-3.59B+104.02C+37.62A*B (d)Residual Plots for Molecular Weight 分析回归方程模型的残差正态概率图，看到残差点近似的分布呈一直线，正负拟合的情况也比较好，说明模型的假设是非常合适的，验证了我得到的统计结果是很有效的。当然图中显示出现了两个异常点，对于模型的适用来说出现这个情况是允许的，不影响广泛性。 Residuals from Molecular Weight vs Molecular Weight （e） Factorial Fit: Visc

18、osity versus A, B, C, D Estimated Effects and Coefficients for Viscosity (coded units)Term Effect Coef SE Coef T PConstant 1500.62 7.524 199.46 0.000A 96.38 48.19 7.523 6.41 0.003B 91.29 45.64 7.524 6.07 0.004C 7.56 3.78 7.524 0.50 0.642D -17.39 -8.70 7.523 -1.16 0.312A*B -14.14 -7.07 7.522 -0.94 0.

19、400A*C 11.85 5.92 7.519 0.79 0.475A*D -23.67 -11.84 7.509 -1.58 0.190B*C 9.82 4.91 7.523 0.65 0.549B*D -25.32 -12.66 7.522 -1.68 0.168C*D 13.22 6.61 7.521 0.88 0.429A*B*C 16.95 8.48 7.484 1.13 0.321A*B*D -6.49 -3.24 7.395 -0.44 0.683A*C*D 4.60 2.30 7.160 0.32 0.764B*C*D 20.35 10.17 7.497 1.36 0.246A

20、*B*C*D 6.20 3.10 2.935 1.06 0.351S = 30.0942 R-Sq = 95.79% R-Sq(adj) = 80.01%Analysis of Variance for Viscosity (coded units)Source DF Seq SS Adj SS Adj MS F PMain Effects 4 47824 71930 17982.4 19.86 0.0072-Way Interactions 6 28864 7258 1209.7 1.34 0.4073-Way Interactions 4 4755 3121 780.2 0.86 0.55

21、64-Way Interactions 1 1009 1009 1008.9 1.11 0.351Residual Error 4 3623 3623 905.7 Lack of Fit 1 1454 1454 1453.9 2.01 0.251 Pure Error 3 2169 2169 722.9Total 19 86074从下面因子效应图中可以看出，A,B对viscosity的影响是最为显著的，需要着重分析。 Factorial Fit: Viscosity versus A, B Estimated Effects and Coefficients for Viscosity (co

22、ded units)Term Effect Coef SE Coef T PConstant 1505.53 15.703 95.88 0.000A -11.61 -5.81 5.076 -1.14 0.269B 61.83 30.92 15.227 2.03 0.058S = 63.0455 R-Sq = 21.50% R-Sq(adj) = 12.26%Analysis of Variance for Viscosity (coded units)Source DF Seq SS Adj SS Adj MS F PMain Effects 2 18503 18503 9251.7 2.33

23、 0.128Residual Error 17 67570 67570 3974.7 Lack of Fit 2 52677 52677 26338.3 26.53 0.000 Pure Error 15 14894 14894 992.9Total 19 86074结论：因为Viscosity 显著影响只和A,B有关，所以回归方程肯定是一条直线，是关于A和B的线性方程，所以不会是曲线的情形。回归方程：U = 1505.53 -5.81A + 30.92B 残差分析与模型合适性检验在正态概率图看到残差很紧密的分布，近似呈现一条直线，表明残差很好的服从正态分布；残差关于零线上下分布，拟合的较好，

24、出现的序列也是杂乱随机的，验证了这个回归模型对实验是非常适用的。题目8-4（a）Fractional Factorial Design Factors: 5 Base Design: 5, 8 Resolution: IIIRuns: 8 Replicates: 1 Fraction: 1/4Blocks: 1 Center pts (total): 0Design Generators: D = AB, E = ACAlias StructureI + ABD + ACE + BCDEA + BD + CE + ABCDEB + AD + CDE + ABCEC + AE + BDE + A

25、BCDD + AB + BCE + ACDEE + AC + BCD + ABDEBC + DE + ABE + ACDBE + CD + ABC + ADEFrom above all, we can get Design Generators: D = AB, E = AC Basic DesignGenerated DesignRunABCD=ABE=AC1-1-1-11121-1-1-1-13-11-1-11411-11-15-1-111-161-11-117-111-1-1811111根据6-21题意可以得到下表RunABCDEYieldLables1-1-1-1116de21-1-

26、1-1-19a3-11-1-1135be411-11-150abd5-1-111-118cd61-11-1122ace7-111-1-140bc81111163abcde(a)Factorial Fit: yield versus A, B, C, D, E Estimated Effects and Coefficients for yield (coded units)Term Effect CoefConstant 30.3750A 11.2500 5.6250B 33.2500 16.6250C 10.7500 5.3750D 7.7500 3.8750E 2.2500 1.1250B

27、*C -1.7500 -0.8750B*E 1.7500 0.8750Effects Plot for yield 从各个因子效应的正态概率图中得出，当显著性水平=0.10时，只有主因素B是非常显著的;但是当=0.28时，因子A,B,C都显得比较重要了（由于试验次数很少，这里采用较大的水平来估计）。（b）对于=0.28时，General Linear Model: yield versus A, B, C Factor Type Levels ValuesA fixed 2 -1, 1B fixed 2 -1, 1C fixed 2 -1, 1Analysis of Variance for

28、yield, using Adjusted SS for TestsSource DF Seq SS Adj SS Adj MS F PA 1 253.12 253.13 253.13 7.11 0.056B 1 2211.12 2211.12 2211.12 62.07 0.001C 1 231.12 231.12 231.12 6.49 0.064Error 4 142.50 142.50 35.63Total 7 2837.87S = 5.96867 R-Sq = 94.98% R-Sq(adj) = 91.21%（c）regression equation: Y = 30.375 + 5.625A + 16.625B + 5.375C(d)Residual Plots for yield 根据上面的残差概率图和残差分布图，残差是随机的出现，并符合正态性分布，假设的模型是非常合适的，结果是非常适用有效的。（e）Residuals from yield vs FITS1 （f） A*D的交互作用最为明显了。（h）