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In this paper, estimates of the parameters and the acceleration factor of a new distribution known as the Weibull Population Mean Distribution (WPMD) are explored. We propose the use of the model suggested by Abd-Elrahman [A better alternative to the generalized Bilal distribution: A new model and applications, Int. J. Reliab. Qual. Saf. Eng. 30(6) (2023) 2350027, doi: 10.1142/S0218539323500274 ] for estimating these parameters. To accomplish this, we employ the maximum likelihood estimation method via a Constant-Stress Partially Accelerated Life Test (CSPALT) model. In this study, we have utilized Unified Hybrid (UH) censored data obtained from the WPMD. Confidence intervals for the model parameters and the acceleration factor are constructed based on the approximate Fisher information matrix. The average and Mean Squared Error (MSE) of estimates are computed to evaluate the performance of the proposed method under the CSPALT model. Additionally, a real data set is provided for illustrative purposes, and a numerical analysis of the UH Censoring Scheme (UHCS) is carried out. Finally, concluding remarks are presented. [ABSTRACT FROM AUTHOR]