## 3 Results and Discussion
### Type I Error Analysis
#### 3.1.3 Quality Control #3  
Because it has been demonstrated (see Section 2.1) that data are not normally distributed, tolerances in direct relation to quantiles can be used for quality control. In the QC3 case, the quantiles from the same error distribution function were used, which means that a high acceptance is expected. In this quality control case, the three err*or categories are determined as follows:  
* CoE1: \(E_{i}\in[Q_{25},\ Q_{75}]\) m
* CoE2: \(E_{i}\in[Q_{5},\ Q_{25})\) m OR \(E_{i}\in(Q_{75},\ Q_{95}]\) m
* CoE3: \(E_{i}\in[(-\infty,\ Q_{5})\) m OR \(E_{i}\in(Q_{95},\ \infty)\) m  
Where the quantiles actuating as tolerances are those presented in Table 4.  
The results obtained by the simulation following the process described at the beginning of this section are shown in Table 9.  
For the three study cases and sample sizes, the proportion of times that the null hypothesis is rejected when it actually is true is close to the significance value (\(\alpha\) = 5%) and does not depend on the sampling size. The tolerances used in this base case of QC3 come from the observed distribution (Table 5), so that the distribution being used is its own observed probabilistic model, and therefore its  
\begin{table}
\begin{tabular}{l c c c c} \hline  & & **Infrastructure** & **Urban** & **Natural** \\ \hline \multirow{5}{*}{Sampling size} & 20 & 0.0492 & 0.0000 & 0.0000 \\  & 50 & 0.0582 & 0.0000 & 0.0000 \\  & 100 & 0.0640 & 0.0000 & 0.0000 \\  & 200 & 0.0810 & 0.0000 & 0.0000 \\  & 500 & 0.1001 & 0.0000 & 0.0000 \\ \hline \end{tabular}
\end{table}
Table 8: Proportion of times that the null hypothesis has been rejected for the QC2 (bias case).  
\begin{table}
\begin{tabular}{l c c c c} \hline  & & **Infrastructure** & **Urban** & **Natural** \\ \hline \multirow{5}{*}{Sampling size} & 20 & 0.0401 & 0.0412 & 0.0398 \\  & 50 & 0.0456 & 0.0492 & 0.0412 \\ \cline{1-1}  & 100 & 0.0430 & 0.0470 & 0.0410 \\ \cline{1-1}  & 200 & 0.0395 & 0.0360 & 0.0420 \\ \cline{1-1}  & 500 & 0.0401 & 0.0458 & 0.0469 \\ \hline \end{tabular}
\end{table}
Table 9: Proportion of times that the null hypothesis has been rejected for the QC3 (true case).  
probability of acceptance and the rejection level approximately equals the significance value of the hypothesis test.  
In short, the interpretation of the three controls (QC1, QC2, and QC3) performed to know the type I error when the new proposed method is applied are conclusive: the proposed method allows to control this type error effectively. This is what happens in the case QC # 3, which is based on the determination of tolerances from the observed data (remember that errors do not follow a normal distribution). This situation is very different from those shown in QC1 and QC2 because in these cases, the adopted base model for deriving the tolerances was based on a Gaussian model. The QC1 and QC2 cases are a clear example that tolerances must be well established to be neither too permissive nor too restrictive in relation to the type I error. The best is that the tolerances are based on the behavior of the observed data, not on models that are supposed to follow that data (usually this is not checked). The method allows us to perfectly adjust the risk of the producer (type I error or the significance), which is an appropriate behavior with respect to the data under analysis and to the interest of quality controls. The statistical behavior of the proposed method is as expected; the observed data only show and confirm its applicability to the use case.