## 3 Results
### Tensile Strength Evaluation
A \(p\)-value conveys a lot of information about the weight of evidence against \(H_{o}\) so that a decision-maker can draw conclusions at any significance level. Thus, the \(p\)-value is the smallest level of significance that would lead to the rejection of the null hypothesis \(H_{o}\) with the given data. It is customary to call the test statistic (and the data) significant when the null hypothesis \(H_{o}\) is rejected; therefore, we may consider the \(p\)-value the smallest level at which the data are significant. Once the \(p\)-value is known, the decision maker can determine how significant the data are without the data analyst formally imposing a preselected significance level (in this case, \(\alpha=0.05\)). Please note that a \(p\)-value of approximately zero implies that it is much lower than the significance level (see Table 5). The ANOVA for the regression model using the Taguchi L27 (3'4) array method can be seen in Table 5. This adequately represents what happens in the process by having a \(p<0.05\), and the calculated F is higher than the F distribution in \(1-\alpha=P(F\leq f\alpha,\,4,\,12)\) with \(\alpha=0.05\), indicating that the model rejects the null hypothesis and concludes that the ultimate tensile strength of the alloy is linearly related to four factors: (A) solutionizing temperature, (B) aging temperature, (C) solutionizing time, and (D) aging time. Additionally, we intend to identify which variables have a greater weight in the model and process, where it is observed that the tensile strength model is less affected by the variable aging temperature with a value \(p=0.109\) (\(p>0.05\)) and a calculated F of 2.79, noting that this has a lower weight in the model. Therefore, this variable does not have statistical significance to the ultimate tensile strength value when compared with the \(p\)-value of the other parameters. This observation is supported by the \(p\)-value reported for the aging temperature variable, which was found to be \(p=0.109\). In contrast, the \(p\)-values for the other variables (solutionizing temperature, solutionizing time, and aging time) were all below the significance threshold of 0.05, indicating their statistical significance with respect to the response variable. The linear regression model was fitted using the data collected by the Taguchi array method from the 27 performed experiments (see Table 4). As a result of this process, the resultant model is shown in Equation (11):  
\[\hat{y}=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{4}x_{4} \tag{11}\]  
where the intercept \((\beta_{0})\) and slope \((\beta_{1},\,\,\beta_{2},\,\beta_{3},\,\beta_{4})\) of the line are called regression coefficients, and \(x_{1},x_{2},x_{3},x_{4}\) are the factors (or regressors) involved in the DOE (solutionizing temperature, aging temperature, solutionizing time, and aging time, respectively) affecting the predicted output \((\hat{y})\), i.e., the UTS value. In that case, the ultimate tensile strength (UTS) can be expressed as (12): \begin{table}
\begin{tabular}{c c c c c c} \hline
**Source** & **SS** & **DF** & **MS** & **F\({}_{0}\)** & \(p\)-Value** \\ \hline Regression Model & 23,078.8 & 4 & 5769.7 & 75.20 & \(\approx\)0 \\ Solutionizing temperature & 5017.6 & 1 & 5017.6 & 65.39 & \(\approx\)0 \\
**Aging temperature** & **214.3** & **1** & **214.3** & **2.79** & **0.109** \\ Solutionizing time & 15,500.5 & 1 & 15,500.5 & 202.02 & \(\approx\)0 \\ Aging time & 2346.4 & 1 & 2346.4 & 30.58 & \(\approx\)0 \\ Residual & 1688 & 22 & 76.7 & & \\ Total & 24,766.9 & 26 & & & \\ \hline \end{tabular}
\end{table} 
Table 5: Summary of the ANOVA based on the 27 experiments performed for this work. Observe that the aging temperature (in bold) is not statistically significant in the UTS value. UTS = \(-1155+1.670*\) solutionizing temp. + 3.45 * aging temp.