
Re: Sampling From Finite Population with Replacement
Posted:
Sep 28, 2010 10:31 AM


On Sep 28, 7:50 am, Cagdas Ozgenc <cagdas.ozg...@gmail.com> wrote: > On Sep 26, 3:25 am, Rich Ulrich <rich.ulr...@comcast.net> wrote: > > > > > On Fri, 24 Sep 2010 03:20:50 0700 (PDT), Cagdas Ozgenc > > > <cagdas.ozg...@gmail.com> 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. > > > >Suppose that we have a population of size N generated by random > > >variable Normal(MeanM, StdDevM). Then take samples of size n < N from > > >this population and calculate average (let's call it MeanS). > > > >MeanS = (1/n)*sum of samples > > > >There is no way you can estimate MeanM in an unbiased fashion. > > > Where do you see "bias"? I think you need to check on that word. > > > > You can > > >only estimate population mean (let's call it MeanP) which is not > > >equal to MeanM, the mean of random variable that generated the > > >population. > > > This population mean is the best "unbiased estimate" of the > > generating mean that you can have here. > > > Where do you get the notion that an unbiased estimatore > > has zero error? It is supposed to be zero "on the average". > > > It is convenient for us that in many cases, the easiest unbiased > > estimate of something in particular is smaller than any of > > the biased estimates, as well as being generally convenient. > > > On the other hand, you can divide either by N, (N1) or > > (N+1) to get three different estimates of the variance > > the normal, each of which has its uses. (N1) gives > > unbiased. I think it is (N+1) that gives minimum variance > > for the estimate. > > > >Is my thinking flawed? Or do we always infer about an hypothetical > > >infinite population? > > > If we are doing experimental science that is intended as > > inferential, there is a future that we point to. For those > > cases, there is an infinite population. That's the only case > > that most of us ever need to worry about. > > > When we are predicting the final election returns from > > the 10 p.m. returns that include 50% of the precincts, > > and using previously known patterns, the N is not infinite. > > >  > > Rich Ulrich > > Here is what I am trying to say. Take any statistics book you will > find a statment that starts something like the following: > > "You have a population of size N with elements normally distributed > with Mu and Sigma. If we sample from this population with > replacement..." Then they continue calculating population mean and > variance, and then claim that Expected Value of Sample mean is equal > to Mu. > > The point I read that it gives me the creeps. First of all normal > distribution is a model. Yes sample mean will give an unbiased > estimate of the population mean (which is a population parameter not a > model parameter). But on average it will not be Mu. Sampling with > replacement from a finite population will not give an unbiased > estimation of the model paramaters. Either I am not reading my books > carefully or this issue is somehow swept under the rug. > > In the infinite population case my understanding is that population > parameter and model paratemer will converge. But when we talk about > inference do we ever care about the model parameter?
I don't understand your objection. Have you ever tried it with a population small enough so that you can enumerate all possible samples of a given size? E.g., try the following:
1. Let the population consist of 5 scores: 2, 3, 4, 5, 6
2. Compute the population mean and SD (with N, not n1 in the denominator).
3. Draw all possible samples of n=2 (with replacement) from the populationthere are 25 of them. For each one, compute the sample mean.
4. Compute the mean and SD of the 25 sample means. For the SD, use N=25 in the denominator, because you have the entire population of sample means.
Notice that the mean of the sample means = the population mean; and the SD of the sample means = the population SD over the square root of the sample size.
 Bruce Weaver bweaver@lakeheadu.ca http://sites.google.com/a/lakeheadu.ca/bweaver/Home "When all else fails, RTFM."

