## 3 Experiment Procedure
### Regression Analysis of Experimental Results
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{\(\bar{Y}_{3}\)} & \multicolumn{4}{c}{\(\bar{Y}_{4}\)} \\ \cline{2-11}
**Source** & \multicolumn{3}{c}{**Sum of Squares**} & **df** & **Mean Square** & **F Value** & **\(p\)-Value Prob \(>\) F** & **Sum of Squares** & **df** & **Mean Square** & **\(F\) Value** & **\(p\)-Value Prob \(>\) F** \\ \hline Model & 1.03 & 14 & 0.073 & 65.63 & \(<\)0.0001 & 1.35 & 14 & 0.096 & 82.31 & \(<\)0.0001 \\ \(X_{3}-P\) & 0.53 & 1 & 0.53 & 47.17 & \(<\)0.0001 & 0.77 & 1 & 0.77 & 656.31 & \(<\)0.0001 \\ \(X_{2}-F\) & 0.14 & 1 & 0.14 & 123.37 & \(<\)0.0001 & 0.12 & 1 & 0.12 & 102.20 & \(<\)0.0001 \\ \(X_{3}-V_{5}\) & 0.030 & 1 & 0.030 & 26.54 & 0.0001 & 0.020 & 1 & 0.020 & 17.12 & 0.0009 \\ \(X_{4}-V_{W}\) & 0.049 & 1 & 0.049 & 44.28 & \(<\)0.0001 & 0.049 & 1 & 0.049 & 42.11 & \(<\)0.0001 \\ \(X_{1}X_{2}\) & 0.030 & 1 & 0.030 & 26.67 & 0.0001 & 0.033 & 1 & 0.033 & 28.35 & \(<\)0.0001 \\ \(X_{3}X_{3}\) & 0.016 & 1 & 0.016 & 14.00 & 0.0020 & 0.017 & 1 & 0.017 & 14.80 & 0.0016 \\ \(X_{3}X_{4}\) & 0.004499 & 1 & 0.004489 & 4.02 & 0.0033 & 0.006241 & 1 & 0.006241 & 5.34 & 0.0354 \\ \(X_{3}X_{5}\) & 0.0001563 & 1 & 0.0001563 & 0.14 & 0.7135 & 0.001523 & 1 & 0.00015823 & 0.16 & 0.6984 \\ \(X_{3}X_{4}\) & 0.003422 & 1 & 0.003422 & 3.07 & 0.1003 & 0.0048481 & 1 & 0.043841 & 2.98 & 0.1048 \\ \(X_{3}X_{4}\) & 0.0003240 & 1 & 0.0003240 & 0.29 & 0.5979 & 0.001056 & 1 & 0.001056 & 0.90 & 0.3567 \\ \(X_{1}\) & 0.001689 & 1 & 0.001689 & 1.51 & 0.2374 & 0.0009164 & 1 & 0.0009164 & 0.078 & 0.7832 \\ \(X_{2}\) & 0.038 & 1 & 0.038 & 34.26 & \(<\)0.0001 & 0.050 & 1 & 0.050 & 42.57 & \(<\)0.0001 \\ \(X_{3}^{2}\) & 0.039 & 1 & 0.039 & 35.40 & \(<\)0.0001 & 0.051 & 1 & 0.051 & 43.81 & \(<\)0.0001 \\ \(X_{4}^{2}\) & 0.004578 & 1 & 0.004578 & 4.10 & 0.0610 & 0.008110 & 1 & 0.008110 & 6.94 & 0.0188 \\ Residual & 0.017 & 15 & 0.001116 & 0.001604 & 11.57 & 0.0073 & 0.017 & 10 & 0.001692 & 13.99 & 0.0047 \\ True Error & 0.0006935 & 5 & 0.0001387 & & 0.00036048 & 5 & 0.0001210 & & 0.0001210 & & \\ Cor Total & 1.04 & 29 & & & 1.36 & 29 & & 1.36 & 29 & \\ \hline \multicolumn{10}{c}{SD = 0.033, Mean = 0.21, C.V.\% = 15.76, PRESS = 0.063,} & \multicolumn{3}{c}{SD = 0.034, Mean = 0.28, C.V.\% = 12.22, PRESS = 0.082,} \\ \multicolumn{10}{c}{\(R^{2}\) = 0.9839, Adj \(R^{2}\) = 0.9689, Pred \(R^{2}\) = 0.9397,} & \multicolumn{3}{c}{\(R^{2}\) = 0.9872, Adj \(R^{2}\) = 0.9752, Pred \(R^{2}\) = 0.9395,} \\ \multicolumn{10}{c}{Adep Precision = 35.244} & \multicolumn{3}{c}{Adep Precision = 38.371} \\ \hline \hline \multicolumn{10}{c}{df\(-\)degrees of freedom, CV–coefficient of variation, F–Fisher’s ratio, \(-\)probability.} \\ \hline \end{tabular}
\end{table} 
Table 6: _Cont._The normal plot of residuals (as shown in Figure 6) shows that the errors are distributed normally. Figure 7 also shows each experimental value matches well with its predicted value. After conducting the experiment, the obtained data were modeled and a quadratic equation for comprehensive goal \(\hat{Y}_{1}\) was achieved as follows:  
\[ \begin{array}{ll}\hat{Y}_{1}=&6.34756+0.00124309X_{1}-0.15942X_{2}-1.35614X_{3 }+0.21733X_{4}+0.0000954688X_{1}X_{2}\\ &+0.000349219X_{1}X_{3}-0.000130469X_{1}X_{4}-0.00321250X_{2}X_{3}-0.00283750X_{ 2}X_{4}\\ &+0.00256250X_{3}X_{4}-0.0000100836X_{1}{}^{2}+0.00349860X_{2}{}^{2}+0.088965X_ {3}{}^{2}-0.014035X_{4}{}^{2}\end{array} \]  
Figure 6: Normal plot of residuals. Figure 7: Predicted vs. actual.