## 2 Materials and Methods
### Analytical and Statistical Methods  
Studies on the technological reliability and the efficiency of pollutant removal in the two chosen facilities were performed in 2017-2019. Wastewater samples for analyses were taken in different seasons (spring, summer, autumn, and winter) from (I) the four-chamber of the preliminary settling tank--after mechanical treatment, and (II) the outflow from an activated sludge chamber--after biological treatment (Figure 1).  
During the study, 20 series of analyses were done and 40 wastewater samples were taken, in which parameters such as total suspended solids (TSS), biochemical oxygen demand in 5 days (BOD5), chemical oxygen demand (COD), total nitrogen (TN), total phosphorus (TP), pH, dissolved oxygen (DO), nitrate nitrogen, nitrite nitrogen, and ammonium nitrogen were determined. Sampling, sample transportation, processing, and analyses were completed on the basis of Polish Standards of Wastewater Examination, which are compatible with the American Public Health Association--APHA [44; 45]. The laboratory apparatus used to carry out the analyses was presented in another paper published by [PERSON] et al. [9].  
The obtained measurement data enabled calculating the mean, minimum, and maximum concentration of pollutant values and their standard deviation. The mean concentrations of the analyzed pollutant parameters in the influent and effluent from the WWTPs were used to determine the efficiency of pollutant removal (Figure 2).  
The evaluation of the technological reliability of the WWTPs was carried out using elements of [PERSON]'s reliability theory, which is a useful tool in assessing the risk of exceeding the normative values in wastewater discharged to the receiver [46].  
The Weibull distribution is characterized by the following probability density function [46]:  
\[f\left(x\right)=\frac{c}{b}\cdot\frac{x-\theta}{b}\overset{\left(c-1\right)}{ \cdot}e^{-\left(\frac{x-\theta}{b}\right)^{c}}, \tag{1}\]  
where \(x\) is a variable describing the concentration of a pollution parameter in the treated effluent, \(b\) is a scale parameter, \(c\) is a shape parameter, \(\theta\) is a position parameter, and \(e\) is a constant, assuming \(\theta<x\), \(b>0\), \(c>0\), and \(e=2.71828\).  
A variable specifying the values of basic pollution indicators (TSS, BOD5, COD, TN, and TP) in treated wastewater (\(n=20\)) was analyzed. The analysis consisted of the estimation of the Weibull distribution parameters using the maximum-likelihood method and the verification of the null hypothesis that the analyzed variable could be  
Figure 2: Average efficiency of pollutant removal in two studied household WWTPs.  
described by the Weibull distribution. The null hypothesis was verified with the Hollander-Proschan test at the significance level of 0.05 [36]. Reliability was determined from the cumulative distribution function plotted in the graphs, taking into account the normative values of the indicators specified in the Polish regulations [47] for wastewater discharged from treatment plants of up to 2000 PE (population equivalent): BOD5--40 mg O\({}_{2}\cdot\)dm\({}^{-3}\), COD--150 mg O\({}_{2}\cdot\)dm\({}^{-3}\), TSS--50 mg\(\cdot\)dm\({}^{-3}\), TN--30 mg\(\cdot\)dm\({}^{-3}\), and TP--5 mg\(\cdot\)dm\({}^{-3}\). In the case of TN and TP, the values defined for wastewater discharged into lakes and their tributaries, as well as directly into artificial water reservoirs located in flowing waters, were adopted as standard values [47]. The analysis was carried out using Statistica 13.