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Re: Sampling From Finite Population with Replacement
Posted:
Sep 25, 2010 8:16 PM
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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
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