Re: Unusual formulae for confidence intervals

Bruce Weaver wrote:

Stephen J. Herschkorn wrote:

Just to clarify, I wrote exactly what I meant.


Ray Koopman wrote:

---- snip ----

- 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.


(If you don't want to solve the quadratic yourself, see Mood and Graybill, for example.)



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 circuitously 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.


---- snip ----



That can be read as giving the *usual* way of constructing the confidence interval, *and* the way to get a more conservative estimate, can it not?


No, as I did not explain quite exactly what the authors said. More precisely, they said the standard error for the estimate is sqrt(P_u (1-P_u) / n). Since we do not know P_u (which is the *population* proportion), we would never plug it in to the formula for the confidence interval. Both the book and the instructor told the students to *always* set P_u to 0.5 in the formula. Yes, this gives one a conservative interval estimate, but nowhere else have I seen such a proscription, which I find unnecessarily conservative.


--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan

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