>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, (N-1) or (N+1) to get three different estimates of the variance the normal, each of which has its uses. (N-1) 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.