Decide if subgroup increases imply overall increase

Suppose the average daily time spent on TikTok rises within each gender subgroup (male and female). Does the overall average have to rise as well? In your answer: (1) give a formal statement and either a proof or a concrete numerical counterexample that demonstrates Simpson's paradox; (2) state necessary and sufficient conditions, in terms of subgroup means and population weights, under which the overall average is guaranteed to increase; (3) discuss how composition shifts (for example, a change in the male share) affect the conclusion; (4) describe a diagnostic you would run on real data to avoid incorrect inferences.

Overview: This question tests grasp of aggregation effects (Simpson's paradox), weighted averages, subgroup composition, and the ability to reason about statistical inference and diagnostic checks across strata.