From Theory to Practice: Exploring the Chi-Squared Goodness-of-Fit Test for Poisson Distributions with Car Parking Data

Abstract This study investigates the impact of design decisions on the outcomes of chi-squared tests when applied to Poisson distributions, using the practical example of arrivals in a parking garage. Through a combination of experiments and simulations, we found that variations in modeling parameters, such as interval length, observation period, and the number of classes, significantly influence the p -values of the chi-squared test. Specifically, increasing interval lengths resulted in reduced sample sizes and greater variance in the average number of events, $$\lambda$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> , leading to lower p -values and contrasting previous findings from the literature. Furthermore, we find that biases, such as those arising from disruptions like COVID-19 lockdowns, have a smaller impact than anticipated, whereas sample size is a dominant factor, with larger samples consistently producing lower p -values. Our results highlight the susceptibility of chi-squared tests to parameter choices, emphasizing the risk of misleading conclusions when testing the fit of a Poisson distribution. We recommend that practitioners carefully select model parameters to avoid false claims of statistical significance and urge reviewers to critically assess test setups.