Explain and resolve Simpson’s paradox

Explain what Simpson's paradox is, then build a specific numerical example in which the treatment has a higher success rate than control within every subgroup, yet the overall rate favors control because the subgroup sizes are uneven. Show the algebra that demonstrates how a weighted average can flip the direction, connect this to confounding and causal graph structure, and list practical fixes such as stratification, standardization, and regression with suitable covariate adjustment. Also explain when the aggregate estimate is the correct target of inference and when you must condition on subgroups instead.

Overview: This question tests grasp of Simpson's paradox, confounding, and causal inference, specifically whether the candidate can interpret subgroup-level versus aggregate numbers and recognize when pooled averages are misleading.