In order to justify his style of teaching, Professor X is trying to evaluate whether or not students who take his Math 101 course are more likely to continue to the next semester than students who take Math 101 from different professors. So the test group is "Professor X's Math 101 Students" and the control group is "Other Professors' Math 101 Students". The dependent variable is whether or not the student continued to the next semester.

I feel like there are two different "populations" we can talk about. The first is the population of students who have already taken the courses. I think we would use a chi-square test to test the hypothesis that there was a significant difference in outcomes based on which Math 101 course the student took. But it also seems like there is a second population: every student who might ever take Math 101 in the future. Although we are interested in determining whether or not Professor X's Math 101 course had a significant impact on the outcomes of the first population of students, it seems like a slightly different problem to infer whether or not the entire population of students who will ever take Math 101 are likely to be affected by the difference in Math 101 courses. Do I have that right or am I overthinking it?

More to the point and particular problem here, does Professor X need to take a simple random sample of the students in his Math 101 course and other Math 101 courses? or should he just use all of the students that every took a Math 101 course, i.e. the entire "population" of students who have ever taken Math 101? Are there benefits to using a random sample in this case or are we just unnecessarily losing information?