Question

A government agency cited a manufacturing company for discrimination since
55% of males applying for a position were hired, and only 42% of the female
applicants were hired. The company responded that there were two types of
positions open for application, assembly ine and management, and that a
higher percentage of women were hired at each position: 85% to 75% for the
assembly line positions and 20% to 15% for the management positions
Explain what seems to be going on here.

Answer 1

I'm still crunching the numbers on this problem, but at a glance it looks like a case of Simpson's paradox, where the correlation in overall data is the opposite of the correlation when the data is divided into subgroups. This happens when there's a significant difference in how the two groups are divided. In this example, because the management positions have much lower acceptance rates than the assembly line positions, it is likely that more women are applying for the management positions and more men are applying for the assembly line positions. If there are a *lot* of men who applied for the assembly line job, and 75% of those men got the job, then this quantity can be greater than all the women hired combined across the two jobs. After all, even if a lot of women applied for the manager job, only 20% of those women got hired, and furthermore, that means few women applied for the assembly line job, so even though 85% of those women got the assembly line job, it's 85% of a small number.

Answer 2

if I can illustrate more simply,

the men: (75% of a lot of men) + (15% of a few men)

vs.

the women: (80% of a few women) + (20% of a lot of women)

you can see how the total # of men could potentially end up being greater based on this distribution