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Re: [ap-stat] "Significant" difference of proportions
Posted:
Nov 3, 2009 4:05 PM
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On Tue, Nov 3, 2009 at 9:02 AM, Robert W. Hayden <hayden@mv.mv.com> wrote:
> Can't help you to define "significant" but the usual inference
> procedures are used all the time with observational studies such as
> this. Ask Chris. O.;-) Salary sex discrimination cases come to mind.
> Rejecting the null essentially tells you that the difference you see
> is more than would be expected were random year-to-year fluctuations
> the only thing going on. Unlike a
> well-designed experiment, it does not provide any clue as to what
> might have caused the change if there is one.
Just a small quibble with that what Robert stated here...
In terms of sampling error associated with a statistic, a census
(collecting data from every element of the population) is better than
a sample. Sampling error is associated with a sample statistic
precisely as a result of randomly choosing some elements and not
others and is the entire basis for the tests taught in the AP
curriculum. If there is no sampling, there is no sampling error.
If you calculate a statistic based on the values of every element in
the population, you have calculated the population parameter--of which
a sample statistic would be an estimate with sampling error. There
are any number of other reasons to prefer a sample, but all else being
equal the census is better. There is no need to generalize to a wider
population (and with uncertainty due to sampling), the population
parameter would be known (with no uncertainly due to sampling). I.e.,
calculation of a mean from a population gives you Mu rather than Xbar.
That said, Robert is right that these tests are used often when no
random sampling is done--notably in the context of experiments and in
any number of types of "observational studies" where no random
*selection* takes place. The justification is that these tests, which
are based on sampling distributions also happen to be good
*approximations* (i.e., the probabilities are not exact) of
randomization or permutation tests. Just like taking every possible
random sample from a population, we can imagine creating every
possible allocation of subjects across groups and use that for our
probability model. It is this (informal) connection between
parametric, sampling based tests and randomization tests that allow us
to make sense of sampling-based tests outside of the context of random
sampling. Sampling based tests serve as an approximate model of
chance as we assess "that the difference you see is more than would be
expected were random year-to-year fluctuations the only thing going
on.
I know its not an AP topic, but student will undoubtedly notice this
bit of slight of hand when we use examples--including experiments
where random selection is not used. (Note the very important
distinction between random selection and random assignment).
Frequently asked questions (FAQ) http://www.mrderksen.com/faq.htm
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