## 3 Materials and Methods
To check the significance of structural parameters b\({}_{0}\), b\({}_{1}\),..., b\({}_{\text{n}}\), the \(t\)-test was used, where the \(t\) statistic has a \(t\)-test distribution with \(n\)-\(k\)-1 degrees of freedom. Null and alternative hypotheses are:  
**H\({}_{0}\):**: \(b_{i}\) = 0 (no linear relationship);
**H\({}_{1}\):**: \(b_{i}\) \(\
eq\) 0 (linear relationship does exist). The critical region is two-sided with the critical value that we read from the tables, the \(t\)-test distribution for a fixed level of significance, and \(n\)-\(k\)-1 degrees of freedom. If the value of \(t\) is in the critical region (the calculated value of \(t\) \(>\) value of \(t\) from a table), then we have to reject H\({}_{0}\) in favor of H\({}_{1}\). Otherwise, there is no basis to reject H\({}_{0}\). In the case of structural parameters from Table 1, all values of \(t\) are in the critical region; thus, a relationship does exist. The regression model for \(y\) (average stocks in the distribution centers) is as follows:  
\[y=-10221.2327311+0.0027863344633x_{1}+16.6820813869x_{2}+131.355655244x_{3}+98 65.9183131x_{4}.\]  
A similar situation can be seen in Table 2, which presents significance tests along with effect sizes for the second case of the analyzed type of demand modeling, the Gamma distribution. In this case, regression analysis indicated that the set of all variables x\({}_{1}\), x\({}_{2}\), x\({}_{3}\), and x\({}_{4}\) is statistically significant. From Table 2, one can observe that all variables are statistically significant; the \(p\)-values are distinctly smaller than 0.05. To check the significance of structural parameters in the table, there are also \(t\)-test values; in each case, they are in the critical region; thus a relationship does exist. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(t\)**-Test** & \(p\)**-Value** \\ \hline x\({}_{1}\)—Yearly sales & 10.35361 & \textless{}0.00001 \\ \hline x\({}_{2}\)—Standard deviation of average sales & & \\ in one warehouse & & \\ \hline x\({}_{3}\)—No. of distribution centers & 10.6077 & \textless{}0.00001 \\ \hline x\({}_{4}\)—Customer service level in distribution centers & & \\ \hline \end{tabular} Source: Own calculations.  
\end{table} 
Table 2: Significance tests for independent variables in a regression model for the Gamma distribution. The regression model for y (average stocks in the distribution centers) in the case of the Gamma distribution is as follows:  
\[\text{y}=-55738.8618951+0.00629264047071\text{x}_{1}+7.08365631632\text{x}_{2}+7 95.996410504\text{x}_{3}+54119.5613891\text{x}_{4}.\]  
Furthermore, model verification was carried out to check whether the econometric models were acceptable. The verification process included:  
* The coefficient of determination, \(R^{2}\), is interpreted as the part of the variance of the dependent variable that can be explained by the variance of the independent variable;
* The properties of residuals (random components) of the regression model: conformity to normal distribution, occurrence of autocorrelation of model residuals (Durbin-Watson test);
* The analysis of variance (\(F\)-statistic at low \(p\)-value). The analysis allows the determination of the uncertainty associated with the predicted values of the variable, which may deviate from the average the higher the variance is. The variance as the mean square deviation of the value of a random variable from its mean value is a measure of dispersion of possible values of the variable and is an indispensable tool for testing the significance of the whole equation [49]. The statistical analyses were performed using the STATISTICA package (StatSoft), assuming a significance level of 0.05. The outcomes are presented in Table 3.  
It can be seen from Table 3 that the \(R^{2}\) parameter shows that the model using the Gaussian distribution explains more than 90% of the variation in the average stocks in the distribution centers, while in the case of the Gamma distribution, it allows explaining about 84% of the total variance.