In a software company there are 2 departments, called A and B. In department A, the fraction of senior programmers (of out the junior programmers in this department) finishing the jobs in time is higher than the fraction of intern finishing the job in time (of out the interns in this apartment).
Same thing happens at department B, the fraction of senior programmers finishing the jobs in time is higher than the fraction of intern finishing the job in time.
is it necessary true that the fraction of senior programmers in either department A or B (out of all senior programmers in either department) is higher than the fraction of intern in either department A or B (out all of interns in either department)?
Provide a counter example (number of senior programmers and interns in each department, and the fraction that finishes the job in time)
REMARK: I don't really think this problem is true or cannot be solved by providing a counter example, but I may be wrong. Please give me some ideas on this problem. Thank you.
This is a question demonstrating Simpson's Paradox.
Yes, it is possible. Counter example: suppose that in department A, 81% (81 out of 100) of senior programmers finish their jobs and 80.5% (161/200) of the interns do. Suppose that in department B, 60% (120/200) of senior programmers finish their jobs and 59.5% (59/100) of interns do.
Overall, this means that only 67% (201/300) of senior programmers finish their jobs, while 73.3% (220/300) of interns do, even though within each department, a greater fraction of senior programmers finish their jobs than interns.
This effect seems paradoxical. The reason it is true is because taking the "overall" average (combining the two departments) hides the true efficiency of the senior programmers. In departments A and B, the senior programmers are more efficient than the interns. Notably, both groups were much less efficient in department B. Most importantly, there are way more interns in department A than B, and vice-versa for the seniors. This means that the excess of seniors in department B (with lower efficiency) brings down the overall senior efficiency, while the surplus of interns in department A (with higher efficiency) brings up the interns' overall efficiency. This makes it appear that the interns are more efficient, while in fact, they are not.