The problem involves comparing the recovery times of two independent groups of patients treated with Drug X and Drug Y, and the objective is to determine whether there is a statistically significant difference between the two treatments using the Mann–Whitney U test, interpreted within the broader frameworks of classical and Bayesian inference. The null hypothesis in this context states that there is no difference between the distributions of recovery times for the two drugs. In other words, the probability that a randomly selected patient from Drug X has a longer recovery time than a randomly selected patient from Drug Y is equal to the probability of the reverse. Symbolically, this can be expressed as (H_0: F_X = F_Y), where (F_X) and (F_Y) denote the distributions of recovery times for Drug X and Drug Y respectively. The alternative hypothesis states that the two distributions are not equal, implying that one drug tends to produce systematically shorter or longer recovery times than the other. This can be written as (H_1: F_X \neq F_Y), reflecting a two-tailed test. The Mann–Whitney U test is appropriate for this data for several important reasons. First, the sample sizes are small, with only five observations in each group, making it difficult to reliably assess normality assumptions required for parametric tests such as the independent samples t-test. Second, the data are measured on at least an ordinal scale, as recovery times can be ranked meaningfully. Third, the Mann–Whitney test does not require the assumption of normal distribution; instead, it is