Re: Unusual formulae for confidence intervals
- From: "Reef Fish" <large_nassua_grouper@xxxxxxxxx>
- Date: 23 Nov 2006 19:18:30 -0800
Greg Heath wrote:
Stephen J. Herschkorn wrote:
Just to clarify, I wrote exactly what I meant.
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.
No, they explicity said to use 0.5 / sqrt(n) for all confidence
intervals for the proportion. Actually, more precisely, they
cicuitously said use sqrt(P_u (1-P_u) / n), where P_u is the population
proportion, and set P_u = 0.5 for a conservative estimate. I am
familiar with the standard techniques for determining sample size; the
book was not discussing this issue.
The reasons are:
1. 0<= Q(1-Q) <= 0.25 for 0<= Q <=1 with the max
occuring at Q = 0.5. Therefore
It is true for ALL Q on the real line.
2. It is easy to remember.
Why do you have to remember it?
The latter is used when the standard deviation is known.
The former is used when it is obtained from the sample
estimate of the variance.
Regardless what you call the former or latter, your statement
is at best misleading when you did NOT specify which form
of the "sample estimate" of the unknown population variance
is used. That was a VALID point made by Koopman about
the ambiguous usage of S that probably led to some of the
present confusion about (n-1) and n, because S/sqrt(n-1)
and S/sqrt(n) could be IDENTICAL if different estimates were
used for S.
If S is estimated from the sample variance use sqrt(n-1) and t
with nu = n-2 degrees of freedom.
That is a trivial statistical result, relevant only to PART of the
problem raised.
Hope this helps.
Greg
Helped yourself, perhaps. I doubt if it helped anyone else,
because you are just rehashing everything that had been said,
and said better, by others, because you came late into the
discussion and NOT fully aware of all the statistical issues
involved.
-- Reef Fish Bob.
.
- References:
- Unusual formulae for confidence intervals
- From: Stephen J. Herschkorn
- Re: Unusual formulae for confidence intervals
- From: Ray Koopman
- Re: Unusual formulae for confidence intervals
- From: Stephen J. Herschkorn
- Re: Unusual formulae for confidence intervals
- From: Greg Heath
- Unusual formulae for confidence intervals
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