實驗設(shè)計分析第六版(美 道格拉斯-蒙哥馬利)課后作業(yè)4

1、實驗設(shè)計分析作業(yè)4 機械電子工程 Home Work #4 (Class note 105, 106)1. Montgomery Problem 6-182. Montgomery Problem 6-26 3. Montgomery Problem 8-4英文版題目6-18（a）Factorial Fit: Etch rate versus A極距, B壓強, C氣流, D功率 Estimated Effects and Coefficients for Etch rate (coded units)Term Effect CoefConstant 776.06A極距 -101.62 -50

2、.81B壓強 -1.62 -0.81C氣流 7.37 3.69D功率 306.13 153.06A極距*B壓強 -7.88 -3.94A極距*C氣流 -24.88 -12.44A極距*D功率 -153.63 -76.81B壓強*C氣流 -43.87 -21.94B壓強*D功率 -0.63 -0.31C氣流*D功率 -2.13 -1.06A極距*B壓強*C氣流 -15.63 -7.81A極距*B壓強*D功率 4.12 2.06A極距*C氣流*D功率 5.62 2.81B壓強*C氣流*D功率 -25.38 -12.69A極距*B壓強*C氣流*D功率 -40.13 -20.06Effects Plo

3、t 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 結(jié)論：從主因子效應(yīng)圖看出A,D因子的效應(yīng)比較顯著；從因子交互作用看出，A*D的交互作用比其他的任何項的交互效應(yīng)都大得多。（b）General Linear Model: Etch rate versus A極距, D功率 Factor

4、 Type Levels ValuesA極距 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極距 1 41311 41311 41311 23.77 0.000D功率 1 374850 374850 374850 215.66 0.000A極距*D功率 1 94403 94403 94403 54.31 0.000Error 12 20858 20858 1738Total 15 531

5、421S = 41.6911 R-Sq = 96.08% R-Sq(adj) = 95.09%(c)Factorial Fit: Etch rate versus A極距, 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極距 -101.62 -50.81 10.42 -4.88 0.000D功率 306.12 153.06 10.42 14.69 0.000A極距*D功率 -153.62

6、 -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 20858 1738 Pure Error 12 20

7、858 20858 1738Total 15 531421So we get Regression: Y = 776.06 - 50.81A + 153.06D - 76.81A*D(d)Residual Plots for Etch rate 殘差分析：從正態(tài)概率圖中，殘差緊密的分布在直線的兩側(cè)，呈線性分布，很好的吻合了正態(tài)性分布假定；殘差關(guān)于擬合值也是上下基本大致對稱的分布的，證明回歸模型的方程結(jié)論是非常符合這個實驗數(shù)據(jù)統(tǒng)計理論的。（e）根據(jù)前面的分析可知，A,D的影響效應(yīng)是B,C的好幾倍，B,C的影響基本可以忽略了。剩余的兩個因子A,D是非常重要的，變成重復(fù)22=4次.但要注意實驗次序應(yīng)

8、重新隨機化排列。A極距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極距, D功率 Factor Type Levels ValuesA極距 fixed 2 -1, 1D功率 fixed 2 -1, 1Analysis of Variance for Etch rate, using Adjusted SS for TestsSource

9、DF Seq SS Adj SS Adj MS F PA極距 1 41311 41311 41311 23.77 0.000D功率 1 374850 374850 374850 215.66 0.000A極距*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 表明當(dāng)A低水平，D高水平時，二者的交互作用非常明顯。(g

10、)Residuals vs Order for Etch rate 我發(fā)現(xiàn)殘差有峰有谷近似沿著某條正弦曲線分布的趨勢。題目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 Molecular Weight (coded units)Te

11、rm 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 11.575 -0.72 0.512B*C -22.3

12、7 -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 = 46.3860 R-Sq = 97.17% R-Sq(

13、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 2437 2436.8 1.13 0.347Re

14、sidual 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 結(jié)論：從上面的分析中看出A、C、A*B的P值均小于0.05，故A、C、A*B對Molecular Weight的影響最大。因為有A,B的交互作用明顯，所以暗示了回歸模型的方程不是線性的，是曲線形狀的。(c)Factorial Fit: Molecular Weight versus A, B, C Estimated Effects and Coefficients

15、 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%Analysis of Variance for Mol

16、ecular 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 303745Regression Equation

17、: Y = 2505.57+69.33A-3.59B+104.02C+37.62A*B (d)Residual Plots for Molecular Weight 分析回歸方程模型的殘差正態(tài)概率圖，看到殘差點近似的分布呈一直線，正負(fù)擬合的情況也比較好，說明模型的假設(shè)是非常合適的，驗證了我得到的統(tǒng)計結(jié)果是很有效的。當(dāng)然圖中顯示出現(xiàn)了兩個異常點，對于模型的適用來說出現(xiàn)這個情況是允許的，不影響廣泛性。 Residuals from Molecular Weight vs Molecular Weight （e） Factorial Fit: Viscosity versus A, B, C, D

18、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 0.400A*C 11.85 5.92 7.519

19、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*B*C*D 6.20 3.10 2.935 1

20、.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 0.5564-Way Interactions 1 10

21、09 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從下面因子效應(yīng)圖中可以看出，A,B對viscosity的影響是最為顯著的，需要著重分析。 Factorial Fit: Viscosity versus A, B Estimated Effects and Coefficients for Viscosity (coded units)Term Effect Co

22、ef 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 0.128Residual Error 17

23、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結(jié)論：因為Viscosity 顯著影響只和A,B有關(guān)，所以回歸方程肯定是一條直線，是關(guān)于A和B的線性方程，所以不會是曲線的情形?；貧w方程：U = 1505.53 -5.81A + 30.92B 殘差分析與模型合適性檢驗在正態(tài)概率圖看到殘差很緊密的分布，近似呈現(xiàn)一條直線，表明殘差很好的服從正態(tài)分布；殘差關(guān)于零線上下分布，擬合的較好，出現(xiàn)的序列也是雜亂隨機的，驗證了這個回歸模型對實

24、驗是非常適用的。題目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 + ABCDD + AB + BCE + ACDEE

25、+ 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根據(jù)6-21題意可以得到下表RunABCDEYieldLables1-1-1-1116de21-1-1-1-19a3-11-1-1135be411-11-150abd5-1-111-118cd61-11-1122ace7-111-1-140bc81111163abcde(a)Factorial Fit: yield versus A, B, C, D, E Estimated Effects and Coefficients for yield (coded units)Term Effect CoefConstan