On Sep 25, 4:25 pm, 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, (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.
Dividing by N+1 minimizes the expected squared error in the estimated variance.
> >> 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