## 3 Results and Discussion
### Type II Error Analysis  
Now, we will study how the method behaves when the null hypothesis is not true; we will study the power of the statistical test. Since our proposal is specially designed for non-normal data, it is necessary to analyze the sensitivity of this method based on error counting when modifying tolerances. Considering the same situation as for QC3 (based on the observed data), we work with four extensions of the case in which the probability limits have been modified with respect to limits specified in the base case (QC3). The first two extended cases (ExC1 and ExC2) are more restrictive than the base case (QC3), and the last two extended cases (ExC3 and ExC4) are less restrictive than the base case (QC3). The newly proposed control limits of \(L_{j}\) (in meters) appear in Table 10 for \(j=5\%,\;25\%,\;75\%,\;95\%\).  
For infrastructure data, the limits are modified by \(\pm 0.5\) cm and \(\pm 1\) cm in relation to the base case, and for urban and natural data, the limits are modified by \(\pm 2\) cm and \(\pm 4\) cm in relation to the base case. Different values are considered for the infrastructure data (\(0.5\) cm and \(1\) cm) and the urban/natural data (\(2\) cm and \(4\) cm) since the distribution of the error data for the first case is narrower than the other cases. In all the extended cases, the three error categories are determined as follows:  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Limits** & \multicolumn{4}{c}{**Extended Cases**} \\ \cline{3-6}  & & **ExC1** & **ExC2** & **ExC3** & **ExC4** \\ \hline \multirow{4}{*}{Infrastructure} & \(L_{5}\) & \(-0.0473\) & \(-0.0423\) & \(-0.0573\) & \(-0.0623\) \\  & \(L_{25}\) & \(-0.0334\) & \(-0.0284\) & \(-0.0434\) & \(-0.0484\) \\  & \(L_{75}\) & \(-0.0157  
* CoE1: \(E_{i}\in[L_{25},\ L_{75}]\) m
* CoE2: \(E_{i}\in[L_{5},\ L_{25}]\) m OR \(E_{i}\in(L_{75},\ L_{95}]\) m
* CoE3: \(E_{i}\in(-\infty,\ L_{5})\) m OR \(E_{i}\in(L_{95},\ \infty)\) m  
The results of this sensitivity analysis for the three study cases are shown in Figure 5. Additionally, for comparison, the base case for QC3 is included in each data category. The curves are very similar for the three study cases. In the two extended cases (ExC1 and ExC2), which are more restrictive than the base case of QC3, the null hypothesis is widely rejected even with small sample sizes (see the upper curves), whereas in the two extended cases (ExC3 and ExC4), which are less restrictive than the base case of QC3, the null hypothesis is always accepted. The curve of the base case of QC3 has been added, and it can be observed that this curve is almost horizontal, with a value of approximately 5%. This sensitivity analysis clearly shows that the method works well against variations in tolerances (statistical power or type II error). If the tolerances are more restrictive, the null hypothesis is rejected with great probability even for small sample sizes. If the tolerances are more generous, the null hypothesis is always accepted.  
Figure 5: Proportion of times that the null hypothesis is rejected (type II error) for the sensitivity analysis of the 3 rd quality control case: (**a**) infrastructure study case, (**b**) urban study case, and (**c**) natural study case.  
In conclusion, the behavior of the proposed control method is expected for a statistical contrast with respect to type II error. In addition, the example shown can guide users of the new method to test with their data and evaluate the power of the statistical contrast they are applying in their specific case.