On 24/09/2010 6:20 AM, Cagdas Ozgenc wrote: > In statistics text books it is proposed that sampling from a finite > population with replacement is equivalent to sampling from an infinite > population. I find this somewhat misleading. >
I suspect what the book or books are getting at is that the distinction between samples drawn with and without replacement becomes less important as the population size increases. Let N be the population size, and p the probability of being sampled. When one draws a simple random sample *with replacement*, p = 1/N for every member of the population on every draw. When one samples *without replacement*, though, the denominator decreases by 1 after each draw. So p = 1/N on the first draw. For those not yet drawn, p = 1/(N-1) on the second draw, 1/(N-2) on the third draw, etc. Once a member has been drawn, p = 0 thereafter.
When the population size is large enough, reducing the denominator by 1 each time has a tiny impact on the probability of being drawn; and likewise the probability of being drawn at any point is very close to 0. So, when the sample size is large enough, the distinction between sampling with and without replacement becomes a pretty fine one. Here are some typical notes on all of this: