You are asking about the so-called finite population correction factor (FPC for short). It applies to the sampling variance of any mean, and therefore to a proportion (which is the mean of an indicator variable). If applied to the equation you cite, one would have SD(p^) = root[(p^(1-p^))/n] root[FPC]. IIRC, FPC = (N-n)/(N-1), or perhaps FPC = (N-n-1)/(N-1). In any case, under the conditions cited in your textbook, FPC = 1, at least approximately, and is therefore negligible in its effect on SD(p^). Reminder: the conditions are "when sampling with replacement" (in which case N is infinite and FPC = 1) or when n << N (in which case, even though N be finite, FPC is not distinguishable from 1 within the precision of the data).
There are few occasions where N is finite, known, and small enough to influence the sampling variance of interest. (And in a goodly proportion of such occasions, upon reflection one discovers that the actual population size is considerably larger than one had credited it to be. For example, in considering the population of Berkeley, CA, is one interested in all the persons who are resident in Berkeley as of today, January 31, 2004; or all the persons who might be properly residents of Berkeley, some of whom are away today; or all the persons who might have been residents of Berkeley today had the circumstances of their lives been different?) When N is unknown, it is often unknown precisely because it is effectively infinite. Hence FPC, which is by and large the only form in which N affects variance estimates, is not ordinarily invoked in practice, and does not appear in elementary textbooks. This may be thought to reflect a somewhat broader truth: the precision of a statistical estimate is a strong function of the sample size, but is not influenced much (sometimes not at all) by the population size. On Fri, 30 Jan 2004, Xiao Li wrote in part: > Let N be the population size. Let p denote the proportion of this > population with a certain characteristic ... [snip] > > Now, my statistics textbook says the standard of deviation is given by > SD(p^) = root[(p^(1-p^))/n] when sampling with replacement or when > the sample size is significantly smaller than the population size. > > My question is why isn't there a N variable in our equation for > SD(p^)? [snip, the rest] ----------------------------------------------------------------------- Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
