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Topic: Sampling From Finite Population with Replacement
Replies: 28   Last Post: Sep 30, 2010 6:30 AM

 Messages: [ Previous | Next ]
 Bruce Weaver Posts: 753 Registered: 12/18/04
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, (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.

>
> > --
> > 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 n-1 in the
denominator).

3. Draw all possible samples of n=2 (with replacement) from the
population--there 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