Re: Unusual formulae for confidence intervals
- From: "Reef Fish" <large_nassua_grouper@xxxxxxxxx>
- Date: 22 Nov 2006 16:14:46 -0800
Ray Koopman wrote:
Stephen J. Herschkorn wrote:
One of my tutees is taking a course in statistics for public policy. I
think this is my first such client. (I have had other studetns in
psychology, sociology, and public health.)
As we were discussing confidence intervals, he pointed out two formulae
in his text and notes. I had never encountered these before.
- For a population proportion, use as the standard error 0.5 /
sqrt(n), where n is the sample size. The justification was that
since we do not know the population proportion pi, use pi = 0.5 for a
conservative interval estimate. Most formulae I have seen substitute
the sample proportion Q (my notation) to get sqrt(Q (1-Q) / n) for
the standard error. If one is going to be a stickler, one can solve the
quadratic inequality
-z <= (Q - pi) / sqrt( pi (1- pi) / n) <= z for explicit bounds on
pi. So using 0.5 seems silly to me.
That's for choosing the sample size (before the data are collected)
so that the standard error is guaranteed to be small enough.
That is only ONE of the reasons for it. A good point about how it's
used for the determination of sample size.
- For a population mean with a large sample size n, use S /
sqrt(n-1), where S is the sample standard deviation, as the standard
error with a normal distribution. I have always seen S / sqrt(n).
Personally, I always use S / sqrt(n) with the t distribution, since
computers can compute t for any degrees of freedom. (Oddly, though
the textbook discusses hypothesis testing with small samples, it does
not discuss confidence intervals with small samples.)
How did they define S^2? With n in the denominator?
Maybe the important part of that recommendation is supposed to be
the substitution of the normal for the t distribution.
That is only superficial reason for addressing the symptom without
addressing the real REASON (if any) in where the usage arises.
Perhaps BOTH are wrong, if the estimation criterion mandated the
use of (n+1) in the denominator for S^2, as one of the three most
commonly used denominators for the estimate of variance from a
sample.
Have you seen these practices elsewhere? Are these conventions peculiar
to public policy?
On another matter, didn't there used to be a newsgroup named sci.stat?
That's where I was going to post this.
And, in case you are wondering, here is the reason I use "Q" for sample
proportion. I prefer to use lower-case Greek letters for parameters and
capital Latin letters for statistics. I rule out "P" for proportion
since "P" is used to mean probability. Hence, I use the next
alphabetical letter.
Whatever works for you.
That's the kind of response that promotes rampant quackery amongst
nonstatisticians practicing statistics on their own whim of
"conventional
use" rather than sound statistical theory and practice.
-- Reef Fish Bob.
--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey
.
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