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Topic: Proportions & Means Tests (fwd)
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Bob Hayden

Posts: 2,384
Registered: 12/6/04
Proportions & Means Tests (fwd)
Posted: Feb 2, 1997 3:02 PM
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----- Forwarded message from -----

Since you do not need the standard dev of the population for inferential
statistics when using proportions my students can do their own real world
experiments without knowing the t-distribution.


----- End of forwarded message from -----

Only if "inferential statistics" = "hypothesis tests". I think many
people would like to put the primary emphasis on estimating a
parameter (with an error estimate called a "confidence interval").
There are statistical reasons for doing this. There is also the fact
that the logic of estimation is much more straightforward than the
logic of hypothesis testing, and it is easier to teach from the
simpler to the more complex.

Inference for proportions involves the binomial distribution, but
using a binomial table for inference is MUCH harder than using a
t-table. If n is large you can approximate the binomial with the
normal distribution (though you can approximate the t-distribution
even better!). To treat binary data this way you have to code
it as 0s and 1s. Your sample will NEVER look normally distributed.
Indeed, most textbooks do not even show the data for proportions, and
tacitly push students toward the worst crime of all: not examining the
data! In these ways the situation with proportions is ATYPICAL of
other inference situations. For that reason, I hesitate to do it
FIRST because then it becomes the paradigm. And what you do in
practice is many more steps away from the theory that underlies what
you are doing. If you explain any of this, there are a lot MORE
things that need explaining. I think these more than offset not
needing to learn to read the t-table. The traditional way around this
is the old distinction between "large sample" and "small sample"
inference. For "large" samples, the t and z values are
indistinguishable, so you can use z even when sigma is unknown. The
problem with that too is that it involves one more level of
approximation, and obscures what is really going on mathematically: t
is correct when sigma is estimated from the sample data, z when sigma
is known, regardless of sample size. You are relying on this
approximation when you use z for proportions; if you are willing to do
that, why not use z for other means? (The tricks for proportions work
only because the proportion is the mean of the 0-1 data, and hence the
theory for means applies to proportions as a special case.)

On the other hand, these are just my opinions, and I am going out on a
limb by disagreeing with someone who has made as large a contribution
to this list as Al Coons has.

| | Robert W. Hayden
| | Department of Mathematics
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| | Plymouth, New Hampshire 03264 USA
| * | Rural Route 1, Box 10
/ | Ashland, NH 03217-9702
| ) (603) 968-9914 (home)
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