# Factorial Design to Stimulate Biomass Development with Chemically Modified Starch
## 3 Results
### Elaboration of the Mathematical Model  
The response function in this study was the biomass development with the variation of some parameters. Three other tests were done in the central point of the domain (0, 0) to calculate the significance of the program:  
\[y_{\text{med}}^{0}=\frac{\sum\limits_{\text{i=1}}^{3}y_{\text{i}}^{0}}{3}=\frac {0.142+0.158+0.079}{3}=0.126 \tag{3}\]  
The average error square was calculated using the following relation:  
\[\varepsilon^{2}=\sum\limits_{\text{i=1}}^{\text{n}}\frac{\left(y_{\text{i}}^{ 0}-y_{\text{med}}^{0}\right)^{2}}{\text{n}-1}=1.744\cdot 10^{-3} \tag{4}\]  
The error for control samples was calculated with relation (5):  
\[\varepsilon=\sqrt{\varepsilon^{2}}=0.042 \tag{5}\]  
Determining the significance of the coefficients was carried out with the following relation, where the number of tests, N, is 27:  
\[\text{S}=\frac{\varepsilon}{\sqrt{\text{N}}}=\frac{0.042}{5.196}=8.038\cdot 1 0^{-3} \tag{6}\]  
The significance of the coefficients was tested with Student's \(t\)-tests using relation (7):  
\[\mathfrak{t}_{\mathfrak{j}}=\big{|}\mathfrak{a}_{\mathfrak{j}}\big{|}/\text{S} \tag{7}\]  
The values of the Student's \(t\)-tests for each coefficient are presented in Table 5.  
Given the results of the Student's \(t\)-tests, it is observed that the following terms can be eliminated: x\({}_{2}\), x\({}_{11}\), x\({}_{123}\), x\({}_{13}\). The mathematical model that describes the function of the response of the criterion of optimization after the elimination of the insignificant terms, with the assistance of the Student's \(t\)-tests, is:  
\[ \begin{array}{c}\text{Y}=0.083+9.889\cdot 10^{-3}.\text{x}_{1}-7.222\cdot 10^{-3}.\text{x}_{3}+0.034\cdot\text{x}_{1}\cdot\text{x}_{2}+0.013\cdot\text{x}_{2} \cdot\text{x}_{3}\\ +9.444\cdot 10^{-3}.\text{x}_{2}^{2}-0.012\cdot\text{x}_{3}^{2}\end{array}  \tag{8}\]  
Further effects of the parameters will be discussed. The value of a\({}_{0}\) (0.083) indicates the fact that we have an optimal quantity of biomass to a value approaching this value. The positive coefficient a\({}_{1}\) means that the variable x\({}_{1}\) has a favorable individual action. The coefficient a\({}_{3}\) is negative; therefore, the variable x\({}_{3}\) has an unfavorable effect on the process. The individual effect of x\({}_{2}\) and the variable x\({}_{2}\) will not be discussed since they were proven to be unsignificant by the Student's \(t\)-tests. The coefficient of interaction a\({}_{12}\) and a\({}_{23}\) is viewed as positive, and the variables x\({}_{12}\) and x\({}_{23}\), by their interaction, have a favorable effect for the process. In analyzing the quadratic coefficients a\({}_{22}\) and a\({}_{33}\), the result is that the function of the answer is characterized by a maximum in connection with the variables x\({}_{1}\) and x\({}_{3}\) and by a minimum in connection with the variable x\({}_{2}\) in which the minimum has a larger curve than the maximum. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(\mathfrak{t}_{\mathfrak{j}}\) & \(\mathfrak{t}_{\mathfrak{0}}\) & \(\mathfrak{t}_{\mathfrak{1}}\) & \(\mathfrak{t}_{\mathfrak{2}}\) & \(\mathfrak{t}_{\mathfrak{3}}\) & \(\mathfrak{t}_{\mathfrak{2}}\) & \(\mathfrak{t}_{\mathfrak{3}}\) & \(\mathfrak{t}_{\mathfrak{2}}\) & \(\mathfrak{t}_{\mathfrak{1}}\) & \(\mathfrak{t}_{\mathfrak{2}}\) & \(\mathfrak{t}_{\mathfrak{3}}\) & \(\mathfrak{t}_{\mathfrak{2}}\) \\ \hline
**Calculated value** & 10.266 & 1.23 & 0.94 & 0.899 & 4.261 & 0.28 & 1.617 & 0.47 & 1.175 & 1.521 & 0.933 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Values of the Student’s \(t\)-tests. For the function of the response obtained after the elimination of the unsignificant terms with the assistance of Student's \(t\)-tests, the drifts partial of order I in connection with each variable were calculated:  
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