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Topic: [ap-stat] Why z for proportions
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Robert W. Hayden

Posts: 4,998
Registered: 12/30/05
[ap-stat] Why z for proportions
Posted: Mar 4, 2011 8:48 AM
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Forwarded message:
> From: Sally Miller <smiller@acps.k12.va.us>

> 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

http://courses.ncssm.edu/math/Stat_Inst/Stats2007/Bob%20Hayden/Relief.html

Robert W. Hayden, P.O Box 450, North Troy, VT 05859
phone (802) 988-2587 web site http://statland.org/
email bob statland.org (add your own "@" and save me some spam)


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