Forwarded message: > From: Sally Miller <firstname.lastname@example.org>
> One of my students asked why we use the z distribution for > proportions? Why not use the t distribution? I assume it is > somehow connect to standard deviations, but am probably completely > wrong?
The reasons may be more historical/pedagogical than mathematical.
For very large samples z works well for both means and proportions. The sampling distribution will be close to normal by the CLT and the error in estimating sigma with s is small.
For means from smaller samples we use t to correct for estimating sigma with s. The SIZE of the correction is based on assuming the POPULATION is normally dsitributed. With proportions, s is dependent on p, so if we hypothesize p then we have hypothesized sigma. For a CI, we use sample data to estiamte sigma. Presumably we could use t but the t correction is based on assuming a normal population. Proportions come from a 0-1 population which cannot be normally distriburted. So t might give the wrong correction here. However, that does not mean it would be worse than z, just that it might not be perfect. AP takes the position that no correction is better than using a correction that might be off. Simulation studies indicate that t actually works better than z, i.e., the t correction, while not perfect, is better than no correction. Minitab has done proportions with t forever. A much better correction is given by Plus 4, though.
AP spends more time than most college courses on the binomial. This gets approximated by the normal, and then that aproximation is used to justify using z for proportions. It makes a nice conceptual link, but the approximation is really not very good. So it's a case of making a choice for pedagogical reasons that is really not an optimal choice in real life. AP is a real outlier in insisting on its way of using z for proportions as being THE right choice. I would say that sudents who use t should get extra credit rather than points off;-) But that would not be a politically correct choice in the context of AP.
-------> First-time AP Stats. teacher? Help is on the way! See