Exercise 2: How Department Choice Produced the Aggregate Gap — Possible Solution
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The mechanism depends on a real pattern: different departments had
genuinely different overall acceptance rates, and women applied
disproportionately to the departments with LOWER acceptance rates
(more competitive fields), while men applied more heavily to
departments with HIGHER acceptance rates (less competitive fields).

Even if every single department admitted men and women at exactly
equal rates within that department - or, as the real data showed, even
admitted women at a higher rate in most departments - the aggregate
admission rate across all applicants is still pulled down for whichever
group applied more heavily to the harder-to-get-into departments. A
woman applying to a highly competitive department faces a genuinely
lower chance of admission than a man applying to a less competitive
one, purely because of which department each applied to - not because
either department treated men and women differently once applications
arrived.

So the aggregate 44.2%-vs-34.6% gap wasn't measuring bias in how
individual departments evaluated candidates - it was measuring the
combined effect of (1) genuinely different acceptance rates across
departments and (2) a genuinely different distribution of which
departments men and women chose to apply to. Combining all departments
into one number erased that real distinction and made it look like a
single, uniform bias when the real pattern was something else
entirely.

ANSWER: Departments had genuinely different overall acceptance rates,
and women applied more heavily to the more competitive (lower-
acceptance-rate) departments while men applied more to less competitive
ones. This alone can produce a lower aggregate admission rate for women
even if every individual department treated male and female applicants
identically (or, as the real data showed, favored women) - the
aggregate number reflects department choice, not per-department bias.

WHY THIS WORKS AS AN ANSWER
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This explains the precise statistical mechanism - different base rates
combined with different application patterns - rather than simply
restating that a reversal occurred.