## 3 Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Regression Summary for Dependent Variable: Print-through C**} \\  & \multicolumn{4}{c}{\(R\) = 0.19670 \(R^{2}\) = 0.03881 Adjusted \(R^{2}\) = —} \\  & \multicolumn{4}{c}{\(F\)(2.37) = 0.74693 \(p<\) 0.48083 Std. Error of Estimate: 4.32250} \\ \hline \multirow{2}{*}{**N = 40**} & \multirow{2}{*}{\(b^{*}\)} & **Std. Err.** & \multirow{2}{*}{\(b\)} & **Std. Err.** & \multirow{2}{*}{_t(37)_} & \multirow{2}{*}{_p-Value_} \\  & & & **of \(b^{*}\)** & & **of \(b\)** & \\ \hline Intercept & & & 12.23850 & 41.73720 & 0.29323 & 0.77099 \\ Density C & 0.00225 & 0.16366 & 0.44678 & 32.43745 & 0.01377 & 0.98908 \\ Penetration C & 0.19659 & 0.16366 & 0.07734 & 0.06438 & 1.20125 & 0.23729 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Linear regression analysis results for cyan prints. Figure 3 shows a 3D surface plot that visualizes the relationship among three variables: the independent variables _Density C_ (X-axis) and _Penetration C_ (Y-axis), and the dependent variable _Print-through C_ (Z-axis). The visible curvature on the surface confirms that the relationship between the variables is not simply linear. A linear regression analysis was also conducted for magenta prints with two independent variables, _Density M_ and _Penetration M_, to assess their influence on the dependent variable _Print-through M_ (Table 6). The correlation coefficient (\(b^{\star}\)) is \(-0.519160\), indicating a moderate negative linear correlation between the independent variables and the dependent variable and suggesting that an increase in the independent variables does not necessarily lead to an increase in the dependent variable. The coefficient of determination \(R\) is 0.51708, suggesting that 51.7% of the variance of the dependent variable (_Print-through M_) can be explained by the independent variables in the model. The adjusted \(R^{2}\), which accounts for the number of predictors in the model, is 0.26737, indicating that the model explains approximately 26.7% of the variability, representing a moderate ability of the model to explain the variability of the dependent variable. Furthermore, the \(F\)-test value of 6.75170 with a \(p\)-value less than 0.00316 indicates that the model is statistically significant, meaning there is less than a 0.316% chance that the results were obtained by chance. The intercept coefficient is 69.89130 with a standard error of 15.93540, while its \(t\)-value is 4.38592 and the \(p\)-value is very small (0.00009), signifying that it is significantly different from 0 at this level of significance, thus denoting a high statistical significance. The regression coefficient (\(b^{\star}\)) for the independent variable _Density M_ is \(-0.51961\), indicating that an increase in this independent variable by one unit decreases the dependent variable by 0.51961 units. With a \(p\)-value of 0.0075, this variable is statistically significant. In \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**SIGN Summary for Dependent Variable: Print-through M**} \\  & \multicolumn{4}{c}{\(R\) = 0.51708 \(R^{2}\) = 0.26737 Adjusted \(R^{2}\)= 0.22777} \\  & \multicolumn{4}{c}{\(F\)**(2.37) = 6.75170 \(p\) \(<\) 0.00316 Std. Error of Estimate: 2.49210**} \\ \hline \multirow{2}{*}{**N = 40**} & \multirow{2}{*}{\(b^{\star}\)} & **Std. Err.** & \multirow{2}{*}{\(b\)} & **Std. Err.** & \multirow{2}{*}{_t(37)_} & \multirow{2}{*}{_p-Value_} \\  & & & **of \(b^{\star}\)** & & **of \(b\)** & \\ \hline Intercept & & & 69.89130 & 15.93540 & 4.38592 & 0.00009 \\ Density M & \(-0.51961\) & 0.14143 & \(-52.74733\) & 14.36986 & \(-3.67069\) & 0.00075 \\ Penetration M & 0.02816 & 0.14143 & 0.00602 & 0.06438 & 0.19913 & 0.84325 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Linear regression analysis results for magenta prints. Figure 3: 3D surface plot of _Print-through C_ against _Density C_ and _Penetration C_.