My study is on the relationship of college persistence of students, perception of students on student affairs services and their academic integration. I was suggested of using mediation analysis. May I know what is the best sample size for a mediation analysis? I currently have 2,338 population.
Larger samples are almost always better, so the best sample size is the entire population. I guess what you mean to ask is "Is this sample large enough?" And I imagine that by mediation analysis, you mean the Baron and Kenny (1986) style of mediation analysis with three variables and a series of regressions. I also imagine that your variables, being things like "college persistence" and "academic integration", are self-reports on discrete scales with lots of variability. So, I expect that 2,338 subjects is quite enough, by which I mean, the results of such an analysis are unlikely to change if you enlarged the sample.
You will notice in the literature on social psychology that mediation analyses are routinely conducted with much smaller samples, on the order of 100 subjects.
Baron, R. M., & Kenny, D. A. (1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173–1182. doi:10.1037/0022-35220.127.116.113
Fritz and MacKinnon (2007) wrote a paper where they described simulations for determining power for different sample sizes, using different methods. You need to know two effect sizes: from X to M, and from M to Y (in Baron and Kenny terminology). Consult table 3 in that paper.