## 6 Experimentation
### Mann-Whitney U Test or Wilcoxon Rank Sum Test  
This test evaluates frequency distributions (the U-statistic) by comparing the medians of Group A with Group B. If the U-statistic for both groups is identical or the \(p\)-value is greater than the critical number used (0.05), the H0 cannot be rejected. The Mann-Whitney test uses Equation (9) to calculate the \(U(x)\) statistic defined as the minimum U for A and B and calculates the value range including the differences of medians for both groups, where n is the size of each group and R(x) is the sum of the position of the ranges of each group (A and B, with x = 1 for Group A and x = 2 for Group B). We ran the rgui (R - graphical user interface) program to conduct the Mann-Whitney U tests [52] using the data from Table 1 for Groups A and B.  
\[U(x)=n1n2+\left(\left(nx(nx+1)\right)/2-\sum\ Rx\right. \tag{9}\]  
Applying Equation (9), the lower value is used to search the Figure 19. Since U1 is the minst value (U1 = \(120-60=65\)), we look for U1 in the Figure 19 and we find that, with a critical value of 0.5, the \(p\)-value is 0.02974. This is less than the \(p\)-value (0.05), thus indicating the probability of rejection of H0. If the critical value used was 0.10, a value of up to 16 would be acceptable for the H0 not be rejected. The Mann-Whitney test was run using two applications--rgui and XLSTAT--because both returned an approximate \(p\)-value value. The results of the one-tailed Mann-Whitney test (Figure 19) can be interpreted as follows: the H0 can be rejected as the \(p\)-value (number denoting the acceptable error possibilities), 0.02974, is less than the critical value (0.05). The resulting data confirm that the population distribution in both groups is different. On this ground, we can infer, according to the applied statistical research criteria, that the model used variables yielding statistically significant results, and H1 holds. This means that the incident resolution times for Group B technicians who used GESTAC were lower than for Group A technicians who did not use the model. The results yielded by 10 Mann-Whitney tests are as follows:  
Wilcoxon signed-rank/test (two-tailed test): \begin{tabular}{l l} \hline \hline V (standard variance) & 2.524 \\ Expected value & 18.000 \\ Variance(V) & 50.875 \\ \(p\)-value & 0.012 \\ alpha & 0.05 \\ \hline \hline \end{tabular}   
Test interpretation:  
H0 = The two samples follow the same distribution  
H1 = The distributions of the two samples are different  
As the computed \(p\)-value is lower than the significance level alpha = 0.05, the null hypothesis H0 should be rejected and the alternative hypothesis H1, accepted. The risk of the null hypothesis H0 being rejected when it is true is less than 1.16 %  
Figure 19: Mann–Whitney test by rgui. If 5 instead of 10 tests are considered, the results are different, as follows: \begin{tabular}{l l} \hline V. & 10 \\ V(standard variance) & 2.524 \\ Expected value & 5.000 \\ Variance(V) & 7.500 \\ \(p\)-value & 0.125 \\ alpha & 0.05 \\ \hline \end{tabular}   
As the computed \(p\)-value is greater than the significance level alpha = 0.05, the null hypothesis H0 cannot be rejected. The risk of the null hypothesis H0 being rejected when it is true is 12.50%. If 7 instead of 10 tests are considered, the results are different, as follows: \begin{tabular}{l l} \hline V. & 15 \\ V(standard variance) & 2.023 \\ Expected value & 7.500 \\ Variance(V) & 13.750 \\ \(p\)-value & 0.043 \\ alpha & 0.05 \\ \hline \end{tabular}   
As the computed \(p\)-value is lower than the \(p\)-value with the significance level alpha = 0.10, the null hypothesis H0 should be rejected, and the alternative hypothesis H1, accepted. The risk the null hypothesis H0 being rejected when it is true is less than 4.31 %. Table 9 summarizes the \(p\)-value variations, depending on number of samples, with a fixed significance level.