## 3 Results
### Estimation of the Capability of Detection and the Capability of Discrimination
That is, the capability of discrimination of this method of analysis in the 85% value is bounded by these values, guaranteeing the measure of fat via NIR spectroscopy with a probability of false non-compliance (\(\alpha\)) and false compliance (\(\beta\)) of 0.05. On the other hand, Figure 6 shows the Operative Curves for a minimum limit. The corresponding test from Equation (1) is applied to determine the minimum amount of protein in flour, in this case, 12% (\(w/w\)). It can be observed that, despite having a minimum of a 12% (\(w/w\)) of protein as a target value, with the PLS calibration model built, only 11.45% (\(w/w\)) can only be assured with probabilities \(\alpha\) and \(\beta\) equal to 0.05. On the contrary, when working with a maximum limit, as in the case of an established maximum value of 5% (\(w/w\)) of fat in milk samples, the applied hypothesis test is the one in Equation (6). Figure 7 shows the Operative Curves for this specific case, concluding that the value that can be obtained to guarantee both probabilities of a false non-compliance and false compliance (\(\alpha\) and \(\beta\)) of 0.05 is 5.22% (\(w/w\)). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Matrix**} & \multirow{2}{*}{**Analyte**} & \multirow{2}{*}{**N**} & \multicolumn{3}{c}{**Range**} & \multirow{2}{*}{**PL = x\({}_{0}\)**} & \multirow{2}{*}{**Y\({}_{\epsilon}\)**} & \multirow{2}{*}{**CC\(\alpha\)**} & \multirow{2}{*}{**CC\(\beta\)**} & \multirow{2}{*}{**CD\(\alpha\)**} & \multirow{2}{*}{**CD\(\beta\)**} \\ \cline{5-8} \cline{8-10}  & & & & **Min** & & & & & & & 84.63, & [84.27, \\  & Fat (\%) & 66 & 81.10 & 86.60 & 85 * & 84.94/85.64 & - & - & 85.36] & 85.73] \\ \multirow{2}{*}{Butter} & Salt (\%) & 66 & 0.00 & 1.20 & 0 & 0.091 & 0.086 & 0.171 & - & - \\  & & & & 1.2 ** & 1.077 & - & - & 1.11 & 1.02 \\ \hline Flour & Protein (\%) & 484 & 9.41 & 14.58 & 12 ** & 11.733 & - & - & 11.73 & 11.45 \\ \multirow{2}{*}{Milk} & Fat (\%) & 192 & 3.65 & 6.16 & 5 ** & 5.079 & - & - & 5.11 & 5.22 \\  & Protein (\%) & 190 & 3.09 & 4.27 & 4 ** & 3.856 & - & - & 3.89 & 3.78 \\ \multirow{2}{*}{Yogurt} & Fat (\%) & 137 & 0.1 & 9.4 & 0 & 0.322 & 0.290 & 0.580 & - & - \\  & Protein (\%) & 142 & 2.8 & 6.4 & 3 ** & 2.867 & - & - & 2.83 & 2.65 \\ \multirow{2}{*}{Olive} & \multirow{2}{*}{Refined olive oil (\%)} & \multirow{2}{*}{76} & \multirow{2}{*}{61} & \multirow{2}{*}{100} & \multirow{2}{*}{80 **} & \multirow{2}{*}{77.30} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{77.03} & \multirow{2}{*}{74.07} \\  & & & & & & & & & & \\ \multirow{2}{*}{Olives} & \multirow{2}{*}{Difufufenician (mg kg\({}^{-1}\))} & \multirow{2}{*}{38} & \multirow{2}{*}{0.00} & \multirow{2}{*}{3.42} & \multirow{2}{*}{0.6 ***} & \multirow{2}{*}{0.901} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{0.90} & \multirow{2}{*}{1.20} \\  & & & & & & & & & & & \\  & & & & & & & & & & & \\  & & & & & & & & & & & \\ \hline \multicolumn{10}{c}{(*) discrimination limit; (**) minimum permitted or established limit; (***) maximum permitted or established limit.} \\ \end{tabular}
\end{table} 
Table 4: Decision limit, capability of detection, and minimum discriminable concentration for each matrix and each analyte obtained for probabilities of a false positive (or false non-compliance) and false negative (or false compliance) \(\alpha=\beta=0.05\) and for r = 3. Figure 5: Capability of discrimination of operative curves at \(\alpha=0.05\) for fat percentage27 in butter samples having 85% as the PL. Lastly, Figure 8 shows the Operative Curves for PL = 0 mg kg\({}^{-1}\) when determining the agrochemical diffufenican in olives. Once again, the hypothesis test applied was also the one in Equation (6), but in this case, the evaluated probabilities are the false positive and the false negative (instead of false non-compliance and false compliance), \(\alpha\) and \(\beta\), respectively, because PL = 0.