## 2 Materials and Methods
### Mathematical Models and Statistical Formulae Applied in Sample Analysis  
Unlike previous research that relies solely on statistical software for hypothesis testing, this study employs a two-method approach, integrating mathematical modeling (Formulas (1)-(4)) to provide a structured theoretical representation of decision-making processes, empirical statistical analysis (IBM SPSS Statistics 27: \(t\)-tests, Chi-Square tests, regression models) to validate findings using observed consumer data, effect size measures (Cohen's d, Cramer's V) to strengthen interpretations beyond conventional significance tests (\(p\)-values). By combining these methodologies, this study increases analytical depth, reduces the reliance on purely empirical findings, and provides a more structured, theory-based framework for understanding consumer decision-making in the smartphone market. The validity of the current research was obtained by the manner in which the study was conducted. The questions were made from similar examples from other research, and also the validity of the content was made by pretesting the questionnaire several times on students [38]. In terms of the reliability of the research, we conducted several pretests of the questionnaire with some students who completed the questionnaire several times, and we had the same answers. Pretesting was also used for the validity of the instrument of measurement. For opinions and perceptions, we used an interval scale and the semantic differential, with values from 1 to 5. Also, on SPSS, we conducted an analysis of the questions regarding the next phone, including some questions about online acquisition. We only had 9 questions like this, and [PERSON]'s Alpha was 0.659, an acceptable value for this number of questions [39]. To identify the factors that influence the likelihood that a young person will buy a smartphone online, we will apply regression analysis, and we will establish a series of hypotheses through these equations:  
\[Y=\beta_{0}+\beta_{1}\times Gender+\beta_{2}\times Income+\beta_{3}\times Favorite \;brand+\beta_{4}\times Maximum\;price+\varepsilon \tag{1}\]  
\(Y\) is the probability of buying smartphones online (1 if bought online, 0 if not);  
\(\beta_{0}\) is the intercept;  
\(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\), \(\beta_{4}\) represents the regression coefficients for the independent variables  
\(\varepsilon\) is the residual error. In this case, if \(\beta_{1}\) > 0, then women are more likely to buy online than men. Also, to understand why a respondent prefers a particular brand, we include a number of independent variables, such as gender, income and online shopping habits, using a multinomial logit model:  
\[\ln(\frac{P(Brand_{i})}{P\Big{(}Brand_{ref}\Big{)}})=\beta_{0}+\beta_{1}\times Gender +\beta_{2}\times Income+\beta_{3}\times Online\; purchase+\beta_{4}\times Maximum\;price \tag{2}\]  
\[P(Brand_{i})\]  
represents the probability that a respondent prefers \(i\) brand;  
\(P\Big{(}Brand_{ref}\Big{)}\) is the probability for the reference brand, for example \"Apple\";  
In this case, if \(\beta_{3}\) > 0, then those who buy online prefer \(i\) brand more than Apple. At the same time, to identify the regression for the maximum price that young buyers will pay for the next handset, we will use the simple regression model:  
\[Y=\beta_{0}+\beta_{1}\times Income+\varepsilon \tag{3}\]  
\(Y\) represents the maximum price that respondents are willing to pay;  
\(\beta_{1}\) is the price sensitivity to income;  
\(\varepsilon\) is the residual error. In this case, \(\beta_{1}\) shows how much income influences the maximum price.