Did the university admit men at a higher rate?

Question

A university with two departments reports admissions by gender.

Department 1 (easy to get into): women admitted 92 of 100 (92%); men admitted 170 of 200 (85%).
Department 2 (hard to get into): women admitted 20 of 200 (10%); men admitted 12 of 150 (8%).

Overall, women were admitted 112 of 300 (about 37%) while men were admitted 182 of 350 (52%), so it looks like men were favored.

Once you account for department, which gender was admitted at the higher rate, and why does the overall rate suggest the opposite?

Accepted Answer

Within every department, women were admitted at the higher rate. Women beat men in Department 1 (92% versus 85%) and in Department 2 (10% versus 8%). So at the level where admission decisions are actually made, women fared better. The pooled rate flips this, a Simpson's paradox caused by a confounder, and there is no sign of bias against women once the confounder is handled. The confounder is which department applicants chose. Department 1 admits almost everyone; Department 2 admits almost no one. Men mostly applied to the easy Department 1 (200 of 350 applicants), while women mostly applied …

Did the university admit men at a higher rate?

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