## 2 Materials and Methods
### Population Sample and Sample Size Determination  
With the purpose of addressing our research question, the first step is to select the sample upon which the study will be conducted. To do so, the municipalities from which data will be collected were initially chosen. It was established as a criterion that these municipalities should be those where data collection would be straightforward, leading to the decision to select municipalities in the north of Spain due to their proximity, specifically those where pneumatic urban waste collection systems coexist with traditional truck collection. Ultimately, the chosen municipalities were San Vicente de Barakaldo and Portugalete, each with two facilities, and Bilbao, Galdakao, and Llodio, each with one facility, totaling 7 facilities serving 56,000 people.  
According to the Spanish National Institute of Statistics (INE), the average number of individuals per household, on 1 January 2023, is 2.48; however, in the Basque Country, the average is 2.36 people per dwelling, slightly lower than the Spanish average. Given that the studied municipalities are located in the Basque Country, the total number of users served by pneumatic urban waste collection is 132,160.  
Based on this average household size of Spanish households provided by the INE, the necessary sample size was then determined to achieve a confidence level of 95%, while maintaining an 8% margin of error. The methodology utilized for this calculation considers the statistical Equation (1) to obtain a required sample within a finite population.  
\[\text{Sample Size}=\frac{\frac{\text{z}^{2}\cdot\text{p}\cdot(1-\text{p})}{ \text{e}^{2}}}{1+\left(\frac{\text{z}^{2}\cdot\text{p}\cdot(1-\text{p})}{\text {e}^{2}\cdot\text{N}}\right)} \tag{1}\]  
where  
z = standard deviation of the normal distribution used to determine the desired confidence level in the study.  
p = probability that the event under study DOES occur.  
1 - p = probability that the studied event will NOT occur (50%).  
e = margin of error assumed in the study representing the uncertainty of the results.  
N = total population size.  
Based on experimentation, the standard deviation value z based on the desired confidence level is shown in Table 1. It can be seen that the z value, which represents the standard deviation of the normal distribution used to determine the desired confidence level in the study, for a confidence level of 95% is 1.96. The probability p that the event under study does occur is 50% and that it does not occur is also 50%. The margin of error e assumed in the study representing the uncertainty of the results is 8% and the population size N is 132,160 users. Therefore, the sample size and the minimum number of surveys to be conducted to obtain data with adequate statistical reliability will be 150 individuals.  
Therefore, to obtain an updated assessment of the magnitude of the variable under study and to verify the correlation between satisfaction with the systems and evaluations on the convenience of their implementation, it was decided to conduct a number of surveys greater than this, targeting users with experience in collection systems, and some of them focused on technicians specialized in the operation of the most representative systems for urban waste management. As the minimum number of surveys required was 150, 166 surveys were carried out. Of this total, 151 surveys were aimed at users with experience in collection systems, and 15 focused on technicians specialized in the operation of the most representative systems for urban waste management.  
\begin{table}
\begin{tabular}{c c} \hline
**Desired Confidence Level** & **Standard Deviation z** \\ \hline
80\% & 1.28 \\
85\% & 1.44 \\
90\% & 1.65 \\
95\% & 1.96 \\
99\% & 2.58 \\ \hline \end{tabular}
\end{table}
Table 1: Standard deviation value based on Desired Confidence Level.