##### Scenario You are an analyst evaluating user comments and model performance on a social-media platform. ##### Question If the probability that a single user comment is positive is p, what is the probability that two independent comments are both positive? Assuming each response is i.i.d., if the first three responses from a user are positive, what is the probability that the fourth response is also positive? Model A returns a positive response in 80 % of cases while Model B returns a positive response in 90 % of cases. At the 5 % significance level, can you conclude Model B is better? Detail the statistical test, null/alternative hypotheses, test statistic and decision rule. ##### Hints Use the multiplication rule for independent events and a two-proportion z-test with a pooled estimate of p.