## 6 Statistical Test  
There are parametric and nonparametric tests to test a statistical hypothesis. The parametric test indicates which sample is better, the non-parametric ones indicate if there is a difference in the samples. In this case, a parametric statistical test is chosen to test which method is the best one for each of the problems presented above.  
To complete the study carried out in this paper, a statistical test is presented in this section to compare the three fuzzy algorithms FBCO, FDE, and FHS, with the two control problems applied in the previous sections. The selected statistical test is Z test and the parameters used are described in Table 14.  
The hypotheses used for the statistical test indicate the following:  
H\({}_{0}\): The proposed Type-1 algorithm without noise and with noise is greater than or equal to the originals algorithm.  
H\({}_{a}\): The proposed Type-1 algorithm without noise and with noise is smaller than the originals algorithm.  
According to the values used in Table 14, there is a rejection zone for values lower than -1.64. The Equation (23) of the z-test is shown as follows:  
\[Z=\frac{\left(\overline{X}_{1}-\overline{X}_{2}\right)-\left(\mu_{1}-\mu_{2} \right)}{\sigma_{\overline{X}_{1}-\overline{X}_{2}}} \tag{23}\]  
Tables 15-17 show the values of Z, and in these tables \"\(S\)\" means that is found evidence of significance and \"\(N.S\)\" refers to which no significant evidence is found. In Tables 15-17 the result of the Evidence of the first row represents the comparison between the proposed Type-1 and the original methods without noise and the result of the evidence on the second row represents the comparison between the proposed type-1 and the original methods with noise.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Controller** & \(\mu_{1}\) & \(\mu_{2}\) & **Z-Value** & **Evidence** \\ \hline \multirow{2}{*}{Water Tank} & FDE FLC without noise & DE without noise & \(-\)0.655 & N.S \\  & FDE FLC with noise & DE with noise & \(-\)24.213 & S \\ \hline \multirow{2}{*}{Inverted Pendulum} & FDE FLC without noise & DE without noise & \(-\)7.525 & S \\  & FDE FLC with noise & DE with noise & \(-\)2.335 & S \\ \hline \hline \end{tabular}
\end{table}
Table 16: Results for the statistical test with Fuzzy differential evolution algorithm.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Level of Confidence & 95\% \\ Alpha & 0.05\% \\ H\({}_{a}\) & \(\mu_{1}<\mu_{2}\) \\ H\({}_{0}\) & \(\mu_{1}\geq\mu_{2}\) \\ Critical Value & \(-\)1.645 \\ \hline \hline \end{tabular}
\end{table}
Table 14: Parameters for the statistical z-test.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Controller** & \(\mu_{1}\) & \(\mu_{2}\) & **Z-Value** & **Evidence** \\ \hline \multirow{2}{*}{Water Tank} & FBCO FLC without noise & BCO without noise & \(-\)2.734 & S \\  & FBCO FLC with noise & BCO with noise & \(-\)0.169 & S \\ \hline \multirow{2}{*}{Inverted Pendulum} & FBCO FLC without noise & BCO without noise & \(-\)1.405 & S \\  & FBCO FLC with noise & BCO with noise & \(-\)2.600 & S \\ \hline \hline \end{tabular}
\end{table}
Table 15: Results for the statistical test with Fuzzy bee colony algorithm.  
The results of the statistical tests show that for most of the cases the proposal used for each of the algorithms is good, however, the important point of this paper is the comparison of the three methods, with respect to the BCO algorithm the six statistical tests have significant evidence, presenting greater problem with the inverted pendulum controller.  
For the case of the DE algorithm of six statistical tests only in five has significant evidence, having problem in the water tank controller, and finally the HS algorithm of six statistical tests four of them have significant evidence, failing both the water tank and inverted pendulum controllers. Statistically we can say that the algorithm BCO is better compared to the DE and HS algorithms.