## 3 Results and Discussion  
The analysis of variance for the random model partition the total variability in the observations into a component that measures the variation between treatments (SS\({}_{\text{Treatments}}\)) and a component that measures the variation within treatments (SSE). In this regard, the hypotheses tested refer to the variance component as it is shown next. \begin{tabular}{l l} H0: \(\sigma_{\text{step width}}^{2}=0\) & H1:\(\sigma_{\text{step width-stylus tip-speed}}^{2}>0\) \\ H0: \(\sigma_{\text{sytus tip}}^{2}=0\) & H1:\(\sigma_{\text{step width}}^{2}>0\) \\ H0: \(\sigma_{\text{speed}}^{2}=0\) & H1: \(\sigma_{\text{sytus tip}}^{2}>0\) \\ H0: \(\sigma_{\text{step width-stylus tip}}^{2}=0\) & H1: \(\sigma_{\text{speed}}^{2}>0\) \\ H0: \(\sigma_{\text{step width-speed}}^{2}=0\) & H1: \(\sigma_{\text{step width-stylus tip}}^{2}>0\) \\ H0: \(\sigma_{\text{sytus tip-speed}}^{2}=0\) & H1: \(\sigma_{\text{step width-speed}}^{2}>0\) \\ H0: \(\sigma_{\text{step width-stylus tip-speed}}^{2}=0\) & H1: \(\sigma_{\text{sytus tip-speed}}^{2}>0\) \\ \end{tabular}   
Before moving to the implementation of the ANOVA for this study, it was first necessary to determine if the ordinary least squares assumptions for all analyzed variables of Equation (1) were being met, i.e., homoscedasticity, normality, and independence. These assumptions are verified through the residual plots that are presented below. As an example, the behavior of the assumptions in the variable parallelism will be explained in detail. The residual histogram is an exploratory tool to prove the general characteristics of the data. Figure 4 shows that the data is not skewed and there are no outliers. The normal probability plot of the residuals for this experiment is shown in Figure 5. The points in this plot represent a straight line that verifies the assumption that the residuals are normally distributed. The residuals in Figure 6 versus fits shows a random order of residuals on both sides of line 0. It has a constant variation without any recognizable patterns in the residual plot. The residuals versus order plot shows all residuals in the order that the data was collected and used to find non-random error, especially of time-related effects. Figure 7 shows that the residuals are not correlated with each other. Figure 4: Residual histogram. Parallelism. Figure 5: Normal probability of residual. Parallelism. In the same way, the fulfillment of the assumptions was verified in the rest of the variables analyzed (angularity, roundness, diameter, and distance); these plots are all shown in Figure 8. Notice, however, that the software does allow adding units to the plots between Figures 4 and 7 given that this is not considered necessary; however, for the sake of comprehension, the authors did want to specify this on the residual versus fits plot. Figure 8: Residual plots. Angularity, roundness, diameter, and distance. Figure 6: Residuals versus fits. Parallelism. Figure 7: Residuals versus order. Parallelism. Subsequently, an ANOVA was applied using the Minitab 16 software to determine whether the predictors or factors are significantly related to the response. The mean square (MS), \(F\)-, and \(p\)-values can be observed in Tables 3 and 4. These tables have been prepared as a summary containing the values obtained for each of the variables studied. For those cases where the \(p\)-value is less than the level of significance \(\alpha\) (0.05), it is concluded that the corresponding hypothesis is significant; that is, this effect is active or does influence the response variable. Similarly, whenever it is concluded that an interaction between factors has a statistically important effect on the response, its interpretation takes precedence over the corresponding main effects, although these are also significant. This is because when the interaction is significant, it governs the behavior of the response in the function of such factors. Under these premises, Tables 3 and 4 show key ANOVA values for all the variables under study.