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mont4e_sm_ch14_supplemental - Supplemental Exercises 14-51...

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14-60 Supplemental Exercises 14-51 a) Estimated Effects and Coefficients for var_1 (coded units) Term Effect Coef SE Coef T P Constant 191.563 1.158 165.49 0.000 factor_A (PH) 5.875 2.937 1.158 2.54 0.026 factor_B (CC) -0.125 -0.062 1.158 -0.05 0.958 factor_A*factor_B 11.625 5.812 1.158 5.02 0.000 Analysis of Variance for var_1 (coded units) Source DF Seq SS Adj SS Adj MS F P Main Effects 2 138.125 138.125 69.06 3.22 0.076 2-Way Interactions 1 540.562 540.562 540.56 25.22 0.000 Residual Error 12 257.250 257.250 21.44 Pure Error 12 257.250 257.250 21.44 Total 15 935.938 The main effect of pH and the interaction of pH and Catalyst Concentration (CC) are significant at the 0.05 level of significance. The model used is viscosity = 191.563 + 2.937x 1 0.062x 2 + 5.812x 12 b) The interaction plot shows that there is a strong interaction. When Factor A is at its low level, the mean response is large at the low level of B and is small at the high level of B. However, when A is at its high level, the results are opposite. c) The plots of the residuals show that the equality of variance assumption is reasonable. However, there is a large gap in the middle of the normal probability plot. Sometimes, this can indicate that there is another variable that has an effect on the response but which is not included in the experiment. For example, in this experiment, note that the replicates in each cell have two pairs of values that are very similar, but there is a rather large difference in the mean values of the two pairs. (Cell 1 has 189 and 192 as one pair and 198 and 199 as the other.) -1 1 185 190 195 200 factor_B Mean Interaction Plot (data means) for var_1 A high A low
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14-61 14-52 a) Factor Type Levels Values Gear Typ fixed 3 20 24 28 Time fixed 2 90 120 Analysis of Variance for flatness, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Gear Typ 2 0.0007591 0.0007591 0.0003796 3.90 0.082 Time 1 0.0024941 0.0024941 0.0024941 25.66 0.002 Gear Typ*Time 2 0.0000505 0.0000505 0.0000253 0.26 0.779 Error 6 0.0005833 0.0005833 0.0000972 Total 11 0.0038870 Term Coef SE Coef T P Constant 0.056500 0.002846 19.85 0.000 Gear Typ 20 -0.011125 0.004025 -2.76 0.033 24 0.007000 0.004025 1.74 0.133 Time 90 -0.014417 0.002846 -5.07 0.002 Gear Typ*Time 20 90 -0.000708 0.004025 -0.18 0.866 24 90 -0.002083 0.004025 -0.52 0.623 There is weak evidence that flatness distortion is different for the different gear types ( p = 0.082). Gear type is significant at α = 0.1, but not at α = 0.05. Also, the gear type 20 coefficient has a p -value = 0.033. Heat- treating time affects the flatness distortion ( p = 0.002). There is no evidence that factors interact ( p = 0.779). 185 190 195 200 -5 0 5 Fitted Value Residual Residuals Versus the Fitted Values (response is var_1) -1 0 1 -5 0 5 factor_B Residual Residuals Versus factor_B (response is var_1) 5 0 -5 2 1 0 -1 -2 Normal Score Residual Normal Probability Plot of the Residuals (response is var_1) -1 0 1 -5 0 5 factor_A Residual Residuals Versus factor_A (response is var_1)
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