## 3 Results, Analysis, and Discussion
### Analysis and Discussion  
A study of the factors of significant influence, regarding the response variable \(Ra\) (\(\upmu\)m), was performed using an analysis of variance. The analysis was completed in several steps, initially, \(Ra\) (\(\upmu\)m) data were directly analyzed, but the values did not follow a normal distribution (Shapiro-Wilk test \(p\)-value \(<\) 0.05) so the analysis could not continue since this requirement was not met. Later, a logarithmic transformation of the data was performed, which maintained the order but smoothed out the effect of outliers (Table 5). The analysis of the transformed data did show a normal distribution of \(Ra\) (\(\upmu\)m) (Shapiro-Wilk test \(p\)-value \(>\) 0.05), shown in Figure 4a. Then, the condition of homoscedasticity was verified (Levene Statistic, \(p\)-value \(>\) 0.05), and the independent data sets had a similar number of cases. Interactions of up to third order were considered in the analysis, and successive iterations were performed until all values were significant. In each iteration, the statistically less significant effect was excluded, in case it had a \(p\)-value higher than 0.05. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Test No.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & _AV_ & _STD_ \\ \hline _LRI1_/LRS1 \(Ra\) (\(\upmu\)m) & 0.35 & 1.06 & 1.27 & 1.03 & 1.22 & 0.61 & 0.73 & 0.98 & 0.91 & 0.30 \\ _LRI1_/LRS2 \(Ra\) (\(\upmu\)m) & 0.81 & 1.23 & 1.31 & 1.33 & 0.19 & 0.73 & 0.48 & 0.62 & 0.84 & 0.39 \\ _LRI2_/LRS1 \(Ra\) (\(\upmu\)m) & 2.94 & 1.31 & 1.2 & 0.13 & 1.24 & **1.88** & 0.87 & 1.66 & 1.40 & 0.76 \\ _LRI2_/LRS2 \(Ra\) (\(\upmu\)m) & 3.00 & 3.95 & 2.11 & 2.91 & 1.30 & 0.51 & 0.33 & 1.70 & 1.98 & 1.18 \\ _LRI3_/LRS2 \(Ra\) (\(\upmu\)m) & 0.48 & 0.64 & 0.56 & 1.13 & 0.34 & 0.53 & 0.46 & 0.59 & 0.59 & 0.22 \\ _LRI3_/LRS2 \(Ra\) (\(\upmu\)m) & 0.59 & 0.56 & 0.78 & 0.92 & 0.67 & 0.63 & 0.42 & 0.53 & 0.64 & 0.15 \\ \(f\) (mm/rev) & 0.05 & 0.05 & 0.10 & 0.10 & 0.05 & 0.05 & 0.10 & 0.10 & & \\ \(N\) (rpm) & 500 & 1200 & 500 & 1200 & 500 & 1200 & 500 & 1200 & & \\ Type of cooling system, \(C\) & \(CCA\) & \(CCA\) & \(CCA\) & \(CCA\) & \(Dry\) & \(Dry\) & \(Dry\) & & & \\ \hline Av & 1.36 & 1.46 & 1.21 & 1.24 & 0.83 & 0.82 & 0.55 & 1.01 & & \\ STD & 1.15 & 1.15 & 0.49 & 0.84 & 0.45 & 0.48 & 0.19 & 0.49 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mg-Al-Mg test. Arithmetical Mean Roughness, \(Ra\) (\(\upmu\)m) in each plate at the entry and at the exit zones of the holes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Source** & **Squares Sum** & **DF \({}^{*}\)** & **Mean Square** & **F** & **Sig.** \\ \hline Corrected model & 11.496 a  & 11 & 1.045 & 3.372 & 0.003 \\ Intercept & 1.486 & 1 & 1.486 & 4.795 & 0.035 \\ \(C\) & 1.882 & 1 & 1.882 & 6.072 & 0.019 \\ \(LRI\) & 4.983 & 2 & 2.491 & 8.038 & 0.001 \\ \(LRS\) & 0.097 & 1 & 0.097 & 0.314 & 0.578 \\ \(C\)\({}^{*}LRI\) & 0.070 & 2 & 0.035 & 0.113 & 0.893 \\ \(C\)\({}^{*}LRS\) & 2.243 & 1 & 2.243 & 7.238 & 0.011 \\ \(LRI\)\({}^{*}LRS\) & 0.471 & 2 & 0.236 & 0.760 & 0.475 \\ \(C\)\({}^{*}LRI\)\({}^{*}LRS\) & 1.749 & 2 & 0.875 & 2.822 & 0.073 \\ Error & 11.158 & 36 & 0.310 & & \\ Total & 24.141 & 48 & & & \\ Corrected total & 22.654 & 47 & & & \\ \hline \hline \end{tabular} \({}^{*}\) DF: degrees of freedom. a R squared = 0.507 (adjusted R squared = 0.357).  
\end{table} 
Table 5: Mg-Al-Mg test. Last ANOVA iteration for Ln\(Ra\) values. From the analysis of variance, it is concluded that the two significant factors in the process are the location regarding the insert, _LRI_, and the type of cooling system, \(C\). These results are consistent with previous works [58,60] and there is also an interaction, unknown so far, between the type of cooling system, \(C\), and the location regarding the specimen, _LRS_. Table 6 shows the percentage of variability attributed to each factor obtained as a percentage of the sum of squares of each significant factor with respect to the total sum of squares of the significant effects.