Modelling students' cognitive achievement skills using the alpha power transformed Lindley probability distribution.

Cognitive achievements in mathematics skill scores are crucial for daily life in modern society. The objectives of this study were to apply the alpha power transformed Lindley probability distribution to students' cognitive achievement skill scores using regression models and to identify the best probability distributions for cognitive achievement, including APTLD, using Young Lives datasets. This study proposes regression modeling using the alpha power transformation Lindley probability distribution for the application of cognitive achievement in mathematics skill scores. The study found that students' average mathematics skill score was 37.01%, with a standard deviation of 14.9, reflecting performance variation. Parental education differed significantly, with 48.5% of mothers and 34% of fathers lacking formal schooling. Additionally, 59% of students lived in rural areas, while 41% resided in urban settings. The average household size was 5.77 members, showing variability in family structures. From the results, the findings show that the mean cognitive achievement in mathematics skill scores (37.01) is greater than the median (33.33), indicating that the data are positively skewed or right-skewed. The APTLD regression model demonstrates the best fit for the data, as indicated by its lowest AIC and BIC values compared to the APTEPLD, TPLD, and TwPLD models. This confirms its superiority in capturing the underlying structure of mathematics skill scores, making it the most suitable model for analyzing cognitive achievement. Therefore, this new model can be considered a significant contribution to the field of statistics and probability methods. Future work on the presented study could extend the APTLD distribution using Bayesian regression models.