In this paper, a new distribution defined on a bounded interval is introduced, and its main properties, such as moments, Lorenz, and Bonferroni curves, are examined. The unknown parameters of the proposed distribution are estimated using several techniques, including maximum likelihood, least squares, weighted least squares, Anderson–Darling, Cramér–von Mises, maximum product spacing, right-tail Anderson–Darling, minimum spacing absolute distance, and minimum spacing absolute-log distance methods. The performance of these estimation methods is evaluated through Monte Carlo simulations under different parameter scenarios. Additionally, a new quantile regression model based on the proposed distribution is developed, offering greater flexibility for modeling bounded dependent variables. The capability of the proposed distribution to represent various hazard rate shapes, such as inverted-bathtub, bathtub, increasing, decreasing, constant, and increasing–decreasing–increasing, is to demonstrate its applicability and flexibility in real data analyses, particularly in cases where traditional models may underperform. Four different real-data applications from the fields of medicine, politics, physics, and education are presented to demonstrate that the proposed model is used as a strong alternative to the well-known Beta and Kumaraswamy distributions in modeling bounded data. The study provides a robust statistical tool for the analysis of bounded data, with potential applications in datasets related to medicine, politics, physics, and educational sciences.