The mean of a random sample of size 81 from a population of size 1 billion
is going to be Normally distributed regardless of the distribution of the
overall population (i.e., the 1 billion). Oftentimes the magic number of
30 is used to say that the mean will have a Normal distribution, although
that is when we're drawing from an infinitely large population. But for
the purposes of determining the distribution of a mean, 1 billion is
effectively infinite. And so, 81 is plenty.
But note that this is for a _random_ sample, not just any kind of sample.
On Fri, 21 Sep 2001, @Home wrote:
> Stan,
>
> Thanks for the detailed explanation. I have one follwoup ?. You say,
>
> "If the original population is normally distributed, the sample means
> will also be normally distributed. Even if the original population
> is skewed, the sample means will still be approximately normally
> distributed given some assumptions, such as that the sample size
> (81) is small compared to the population size (unknown). I don't
> know enough to state all the conditions precisely."
>
> Asssume the facts already given. Suppose the population was some demographic
> aspect of each person living in India, - n = 1 billion. The _mean_ of
> sample means stays at 3. If the one billion population was badly skewed, is
> it possible that a sample size of 81 would NOT result in a normal
> distribution and would require a larger sample size.
>
>
>
> "Stan Brown" <[EMAIL PROTECTED]> wrote in message
> [EMAIL PROTECTED]">news:[EMAIL PROTECTED]...
> > I'm just a journeyman in this area, but I'm going to presume to
> > answer in hopes that if I make any errors the real gurus will
> > correct me and the shame will facilitate my learning. :-)
> >
> > @Home <[EMAIL PROTECTED]> wrote in sci.stat.edu:
> > >I am trying to solve a ? which basically gives the following facts:
> > >population of unknown number
> > >popu std dev of 27
> > >pop mean of 78
> > >sample of size n=81
> > >2000 random samples
> > >
> > >what is the sample mean?
> > >what is the std error (std dev of sample means)
> > >what shape would the histogram be?
> > >
> > >The sample mean is obviously 78 and I calculate the std error of the
> sample
> > >means to be 3.
> >
> > I think you mean the _mean_ of sample means? The mean of one sample
> > could obviously be anything, though we expect it to be within 78-2*3
> > to 78+2*3 about 95% of the time.
> >
> > I calculate the standard error or the mean (sigma-sub-xbar) the same
> > way you do, as sigma/sqrt(n) or 27/9 = 3.
> >
> > >However I can't put the whole picture together. I suspect the distrib
> would
> > >be normal given the 81 samples, but is 3 a low number for a std error.
> >
> > If the original population is normally distributed, the sample means
> > will also be normally distributed. Even if the original population
> > is skewed, the sample means will still be approximately normally
> > distributed given some assumptions, such as that the sample size
> > (81) is small compared to the population size (unknown). I don't
> > know enough to state all the conditions precisely.
> >
> > >Is it possible to translate it into a z score without any addtional data.
> >
> > If the population mean and standard deviation are known, that's all
> > you need for a z score. The formula is
> > z = [ xbar - mu ] / [ SEM ]
> > For your scenario,
> > z = (xbar-78)/3
> >
> > A sample mean of 60 has a z score of -6, so it is quite unlikely
> > that you'd draw a sample with a mean of 60. (My TI-83 says that the
> > area in the tail past z=-6 is just under 10^-9.)
> >
> > >In other words is the std deve of 27 and mean of 81 in any way predictive
> of
> > >what a histogram of a distribution would look like?
> >
> > I assume you meant to say "mean of 78 and sample size of 81"?
> > Assuming that, the histogram of sample means should be normal or
> > nearly so, with mean (mu-sub-xbar) 78 (same as population mean) and
> > standard deviation (standard error of the mean, sigma-sub-xbar) 3.
> >
> > >Finally what difference does it make how many random samples you take
> (ie.
> > >100 or 1000). What statistic or parameter does this speak to?
> >
> > None that I know, in a formal sense. If you take 100 random samples
> > of size 81, or 100,000 random samples of size 81, your histogram of
> > sample means will have the same shape, though the curve will be a
> > bit smoother with 100,000 samples.
> >
> > --
> > Stan Brown, Oak Road Systems, Cortland County, New York, USA
> > http://oakroadsystems.com
> > My reply address is correct as is. The courtesy of providing a correct
> > reply address is more important to me than time spent deleting spam.
>
>
>
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