<p>This study introduces a new bounded probability distribution, termed the unit exponential odd exponentiated distribution, and investigates its important statistical properties. The parameters of the proposed model were estimated using the maximum likelihood estimation (MLE) method, with a simulation study conducted to evaluate the performance of the estimation process. To demonstrate its applicability, the model was applied to two real-world datasets: the human development index and maximum flood level. The analysis employed both classical and Bayesian approaches. The classical approach evaluated various model selection criteria and goodness-of-fit statistics. Results indicated that the proposed model outperformed nine competing models across multiple criteria. The study provides a theoretical derivation and practical implementation of Wald’s Sequential Probability Ratio Test (SPRT). The integration of the SPRT with the proposed distribution effectively captured critical patterns in the datasets, enhancing the model’s utility in real-world scenarios. A unit-interval quantile regression model is also proposed, demonstrating superior fit compared to conventional beta regression and Kumaraswamy regression models for outcomes confined within the unit interval exhibiting skewness. These findings highlight the significant contribution of the proposed model to the field of statistical modeling and data analysis.</p>