## 2 Materials and Methods
### Particularities of the Factorial Designs under Study  
As briefly mentioned before, the factorial experiment presented in this section and its subheadings focused on the analysis of the cold tube drawing process in order to study the potential impact and/or influence of three factors over certain response variables. The design consisted of three fixed factors at two levels each, yielding a total of eight treatments. The factors were defined as (\(A\)) the diameter of the pre-tube (also known as the draft tube), (\(B\)) the wall thickness of the pre-tube, and, finally, (C) the diameter of the die. Given that the authors mostly took into account the necessary financial resources and \begin{table}
\begin{tabular}{c c} \hline \hline
**Null Hypotheses** & **Alternate Hypotheses** \\ \hline H\({}_{0}\): \(\sigma_{1}^{2}=0\) & H\({}_{1}\)\(\sigma_{1}^{2}>0\) \\ H\({}_{0}\): \(\sigma_{6}^{2}=0\) & H\({}_{1}\)\(\sigma_{6}^{2}>0\) \\ H\({}_{0}\): \(\sigma_{7}^{2}=0\) & H\({}_{1}\)\(\sigma_{7}^{2}>0\) \\ H\({}_{0}\): \(\sigma_{7\beta}^{2}=0\) & H\({}_{1}\)\(\sigma_{7\beta}^{2}>0\) \\ H\({}_{0}\): \(\sigma_{7\gamma}^{2}=0\) & H\({}_{1}\)\(\sigma_{7\gamma}^{2}>0\) \\ H\({}_{0}\): \(\sigma_{7\gamma}^{2}=0\) & H\({}_{1}\)\(\sigma_{7\gamma}^{2}>0\) \\ H\({}_{0}\): \(\sigma_{7\beta\gamma}^{2}=0\) & H\({}_{1}\)\(\sigma_{7\beta\gamma}^{2}>0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Formulation of the hypotheses. time, a total of 4 replications were executed for variables \(y_{1}\) and \(y_{3}\), and 3 for the variable \(y_{2}\), which resulted in a total of 32 and 24 runs, leading to a more robust take on the process under study and, of course, meeting the requirement of a minimum of 20 data entries suggested in Minitab for analyzing the normality assumption of the data. On the other hand, since the authors wanted to study the effect of these factors on three concrete separate response variables, it was decided to define three separate \(2^{3}\) factorial designs or cases for these purposes, as shown in Tables 5-8. Notice, however, that Table 8 is common to all three designs, and thus it is only presented once. The response variables under study were denominated as \(y_{r}\), where \(r=\overline{1,o}\), and such variables were defined as the (1) the roundness of the tube, (2) the final outer diameter of the tube, and (3) the final or resulting wall thickness. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \multicolumn{3}{c}{**Coded Factors**} & \multicolumn{3}{c}{\(y_{2}\) (Final Outer Diameter of the Tube in mm)} \\ \hline
**Run** & \(A\) & \(B\) & \(C\) & **Replicate 1** & **Replicate 2** & **Replicate 3** & **Replicate 4** & **Total** \\ \hline
1 & \(-1\) & \(-1\) & \(-1\) & 0.0226715 & 0.0310896 & 0.0295886 & 0.0269921 & 1 = 0.087670 \\
2 & \(-1\) & \(-1\) & 1 & 0.0519339 & 0.0539362 & 0.0516811 & 0.0538788 & \(c\) = 0.159496 \\
3 & \(-1\) & \(1\) & \(-1\) & 0.0465677 & 0.0478089 & 0.04385197 & 0.0457862 & \(b\) = 0.137447 \\
4 & \(-1\) & \(1\) & 1 & 0.03269045 & 0.0355279 & 0.0325575 & 0.0321517 & \(bc\) = 0.100237 \\
5 & \(1\) & \(-1\) & \(-1\) & 0.0782915 & 0.0748141 & 0.0701904 & 0.070804 & \(a\) = 0.215809 \\
6 & \(1\) & \(-1\) & 1 & 0.0942585 & 0.0956257 & 0.0965571 & 0.0921726 & \(ac\) = 0.284355 \\
7 & \(1\) & \(1\) & \(-1\) & 0.0147617 & 0.0150973 & 0.0164774 & 0.0167065 & \(ab\) = 0.048281 \\
8 & \(1\) & \(1\) & 1 & 0.0568767 & 0.0577252 & 0.05673148 & 0.05634873 & \(abc\) = 0.170805 \\ \hline \end{tabular}
\end{table} 
Table 5: General experiment design table for the case of the response variable \(y_{1}\) (the roundness of the tube).