## 3 Results and Discussion
The last plot refers to the residuals versus the order of the data. In this case, the plot shows no trends, patterns, or shifts, and it is possible to appreciate the residuals are randomly located around the center line. In consequence, the authors can assume the residuals are uncorrelated with each other, or that they are independently distributed, which is another ordinary least squares assumption met by the data under question. Figure 6: A closer look into the fulfillment of the ordinary least squares assumptions for the response variable \(y_{1}\) (final roundness of the tube): (**a**) residual plots for \(y_{1}\); (**b**) results of the Anderson–Darling test for normality for \(y_{1}\). Once it was verified that all the ordinary least squares assumptions had been met, it was possible to continue with the ANOVA for all of the studied variables. For these purposes, the software Minitab, version 19, was selected and used. A summary of the key statistics and values of the ANOVA results is shown in Tables 11 and 12. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{\(y_{1}\) (Final Roundness of the Tube)} & \multicolumn{4}{c}{\(y_{2}\) (Final Outer Diameter of the Tube)} \\ \hline
**Source** & **MS** & **F** & \(p\)**-Value** & **MS** & **F** & \(p\)**-Value** \\ \hline Diameter of the pre-tube (\(A\)) & \(3.295\times 10^{-3}\) & \(6.684\times 10^{2}\) & \(\begin{array}{c}P<0.001\times 10^{-13};\\ p\to 0\end{array}\) & \(5.125\times 10^{-7}\) & \(0.70\) & \(\mathbf{0.415}\) \\ Wall thickness (\(B\)) & \(4.675\times 10^{-3}\) & \(9.485\times 10^{2}\) & \(\begin{array}{c}P<0.001\times 10^{-13};\\ p\to 0\end{array}\) & \(5.823\times 10^{-3}\) & \(7.948\times 10^{3}\) & \(\begin{array}{c}P<0.001\times 10^{-13};\\ p\to 0\end{array}\) \\ Diameter of the die (\(C\)) & \(2.796\times 10^{-3}\) & \(5.673\times 10^{2}\) & \(\begin{array}{c}P<0.001\times 10^{-13};\\ p\to 0\end{array}\) & \(2.397\times 10^{-1}\) & \(3.272\times 10^{7}\) & \(\begin{array}{c}P<0.001\times 10^{-13};\\ p\to 0\end{array}\) \\ \(A\ast B\) & \(4.445\times 10^{-3}\) & \(9.017\times 10^{2}\) & \(\begin{array}{c}P<0.001\times 10^{-13};\\ p\to 0\end{array}\) & \(4.532\times 10^{-5}\) & \(6.187\times 10^{1}\) & \(\begin{array}{c}0.069\times 10^{-5};\\ p\to 0\end{array}\) \\ \(A\ast C\) & \(1.239\times 10^{-3}\) & \(2.514\times 10^{2}\) & \(\begin{array}{c}P<0.001\times 10^{-13};\\ p\to 0\end{array}\) & \(4.532\times 10^{-5}\) & \(6.187\times 10^{1}\) & \(\begin{array}{c}0.069\times 10^{-5};\\ p\to 0\end{array}\) \\ \(A\ast C\) & \(1.239\times 10^{-3}\) & \(2.514\times 10^{2}\) & \(\begin{array}{c}P<0.001\times 10^{-13};\\ p\to 0\end{array}\) & \(2.510\times 10^{-4}\) & \(3.836\times 10^{2}\) & \(\begin{array}{c}P<0.001\times 10^{-5};\\ p\to 0\end{array}\) \\ \(B\ast C\) & \(1.622\times 10^{-3}\) & \(0.329\times 10^{2}\) & \(\begin{array}{c}0.0653\times 10^{-4};\\ p\to 0\end{array}\) & \(1.014\times 10^{-3}\) & \(1.384\times 10^{3}\) & \(\begin{array}{c}P<0.001\times 10^{-13};\\ p\to 0\end{array}\) \\ \(A\ast B\ast C\) & \(1.686\times 10^{-3}\) & \(3.421\times 10^{2}\) & \(\begin{array}{c}P<0.001\times 10^{-13};\\ p\to 0\end{array}\) & \(1.801\times 10^{-5}\) & \(2.459\times 10^{1}\) & \(\begin{array}{c}0.014\times 10^{-2};\\ p\to 0\end{array}\) \\ Error & \(4.930\times 10^{-6}\) & & \(7.325\times 10^{-7}\) & & \(\begin{array}{c}P\to 0\end{array}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Summary of key ANOVA values for the response variables \(y_{1}\) and \(y_{2}\). Figure 7: Fulfillment of the ordinary least squares assumptions for the response variables \(y_{1}\) and \(y_{2}\): (**a**) residual plots for the response variable \(y_{2}\) (final diameter of the tubes); (**b**) residual plots for the response variable \(y_{3}\) (final wall thickness of the tube). Based on the \(p\)-values from the previous Tables 11 and 12, and also on the Pareto graphs related to each of the ANOVAs, the authors of this paper can conclude that there are many significant terms affecting the behavior of the response variables under study.