## 3 Results
contrast, the independent variable _Penetration_\(M\) has a regression coefficient (_b*_) of 0.02816 but a _p_-value of 0.84325, suggesting that this variable is not statistically significant and likely does not contribute to the model. Therefore, _Density_\(M\) is a significant predictor, while _Penetration_\(M\) is not significant in this model. Given that the coefficients from the regression analysis have shown significance and there is a moderate \(R^{2}\) value, it can be postulated that there exists some form of non-linear relationship. Therefore, for the visualization of the relationship between the three observed variables, a 3D surface plot has been chosen, specifically a non-linear spline model (Figure 4). The X-axis represents the independent variable _Density_\(M\), the Y-axis represents the independent variable _Penetration_\(M\), and the Z-axis represents the dependent variable _Print-through_\(M\). Based on the surface shape and the color gradations on the graph, it can be deduced that there is some form of non-linear relationship between the observed independent variables and the dependent variable. This implies that a higher density and penetration lead to an increased print-through, indicating that there is an optimal level of _Density_\(M\) at which _Print-through_\(M\) is minimized, and beyond that point, an increase in _Density_\(M\) leads to an increase in _Print-through_\(M\). In the third step, a linear regression analysis was performed for the yellow prints with two independent variables, _Density_\(Y\) and _Penetration_\(Y\), to assess their impact on the dependent variable _Print-through_\(Y\) (Table 7). The correlation coefficient (\(R\)) value of 0.14506 indicates a very weak linear correlation between the independent variables _Density_\(Y\) and _Penetration_\(Y\) and the dependent variable _Print-through_\(Y\). However, given that the R value is very close to zero, it can be concluded that the linear relationship between the independent variables and the dependent variable \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Regression Summary for Dependent Variable: Print-through Y**} \\  & \multicolumn{4}{c}{\(R\) = 0.14506 \(R^{2}\) = 0.02104 Adjusted \(R^{2}\) = —} & \multicolumn{4}{c}{\(F\)(2.37) = 0.39768 \(p\) \(<\) 0.67472 Std. Error of Estimate: 2.42650} \\ \hline \hline
**N = 40** & \(b^{*}\) & \
\begin{tabular}{c} **Std. Err.** \\ **of**\(b^{*}\) \\ \end{tabular} & \(b\) &
\begin{tabular}{c} **Std. Err.** \\ **of**\(b\) \\ \end{tabular} & \(t\)**(37)** & \(p\)**-Value** \\ \hline Intercept & & & \(-5.45168\) & \(13.11423\) & \(-0.41571\) & \(0.68002\) \\ Density Y & 0.14252 & 0.16419 & 14.70130 & 16.93664 & 0.86802 & 0.39098 \\ Penetration Y & 0.01389 & 0.16419 & 0.00281 & 0.03317 & 0.08460 & 0.93303 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Linear regression analysis results for yellow prints. Figure 4: 3D surface plot of _Print-through_\(M\) against _Density_\(M\) and _Penetration_\(M\). is negligible. Looking at the coefficient of determination, or \(R^{2}\), which is 0.02104, it is evident that the model explains only 2.1% of the variability of the dependent variable _Print-through Y_. This is quite low and suggests that the model does not account for a significant portion of the variability of the dependent variable. Since the adjusted \(R^{2}\) is even lower than the \(R^{2}\), it further emphasizes that the independent variables _Density Y_ and _Penetration Y_ do not significantly contribute to explaining the variability of _Print-through Y_. The intercept coefficient is \(-5.45168\), but with a \(p\)-value of 0.68002 and a standard error of 2.42650, this result is not statistically significant. The regression coefficients (\(b^{\star}\)) for the independent variable _Density Y_ of 0.14252 and for _Penetration Y_ of 0.01389 are also not statistically significant, considering their \(p\)-values of 0.39098 and 0.93303, respectively.