A university has two departments, $X$ and $Y$. The following admission numbers are observed:

Explain why the apparent gender disadvantage disappears when conditioning on department type.

Because conditioning on department removes a confounding effect due to different application shares across departments, so overall rates are weighted averages of within-department rates

Because the observed pattern is due to random sampling error and would vanish with larger samples, so department-specific comparisons are unreliable

Because admissions data are mismeasured at the department level, and aggregating hides the measurement error making the overall rate more accurate

Because the interaction between gender and department indicates effect modification, meaning the causal effect of gender changes sign within departments