Department vs University-Wide Admission Rates by Gender: Can They Disagree?

At a university, every department admits female-identifying applicants at a higher rate than all other applicants. Could the university-wide acceptance rate for female-identifying applicants still be lower than the rate for everyone else? Explain your answer. ```hint Look at the weights A university-wide rate pools all departments together. Consider what determines how much each department counts toward one group's pooled rate, and whether that can differ between the two groups. ``` ### Clarifying Questions - Is the acceptance rate counted per applicant or per application, and can one person apply to more than one department? - Is the comparison group everyone who does not identify as female, or male-identifying applicants only? - Does "every department" mean a strictly higher rate in each one, with no ties? ### What a Strong Answer Covers - A clear yes or no, backed by the mechanism that links department rates to the university-wide rate - A small numeric example with correct arithmetic in which one group leads in every department yet trails overall - The conditions under which the reversal can and cannot occur - Which comparison answers which question, such as bias in admission decisions versus overall outcomes, and what extra data would be needed - How the same pattern shows up in product metrics or experiment readouts ### Follow-up Questions - If a board member asks whether the admissions process is biased against women, which numbers would you show, and what could still confound them? - How could the same reversal appear in an A/B test readout, and how would you design the experiment to prevent it? - If the university wanted the university-wide rates to match, which levers could it use, and which of them would be fair?