At 12:49 PM 11/21/01 -0500, Ronny Richardson wrote:
>As I understand it, the Central Limit Theorem (CLT) guarantees that the
>distribution of sample means is normally distributed regardless of the
>distribution of the underlying data as long as the sample size is large
>enough and the population standard deviation is known.
nope ... clt says nothing of the kind
it says that regardless of the shape of the target population ... as n
increases, the shape of the sampling distribution of means is better and
better APPROXIMATED by the normal distribution
that is, even if the target population is quite different from normal ...
if we take decent sized samples ... we can say and not be TOO wrong that
the sampling distribution of means looks something like a normal ...
here is a quick simulation taking samples of n=50 (based on 10000 samples)
from a chi square distribution with 1 df
.
..::..
:::::::::.
.::::::::::::.
.::::::::::::::..
.::::::::::::::::::.
..::::::::::::::::::::::..
.....:::::::::::::::::::::::::::::............ .
+---------+---------+---------+---------+---------+-------C51
0.30 0.60 0.90 1.20 1.50 1.80
even though the chi square distribution is radically + skewed, the sampling
distribution looks pretty darn close to a normal distribution ... but it
never will be exactly one ...
it does NOT say that it will GET to and BECOME a normal distribution
if the population is not normal ... the sampling distribution will not be
normal regardless of n ... but, it could be that your EYES could not tell
the difference
>It seems to me that most statistics books I see over optimistically invoke
>the CLT not when n is over 30 and the population standard deviation is
>known but anytime n is over 30. This seems inappropriate to me or am I
>overlooking something?
you are mixing two metaphors ...
if we know the sd of the population ... then we know the real sampling
error ... ie, standard error of the mean ... if we do NOT know the
population sd, and substitute our estimate of that from the sample, then we
are only estimating the standard error of the mean
thus ... knowing or not knowing the population sd helps us to know or only
to estimate the real standard error ... but this is unconnected with shape
of sampling distribution
shape of sampling distribution is partly a function of shape of population
AND random sample size ...
>When the population standard deviation is not know (which is almost all the
>time) it seems to me that the Student t (t) distribution is more
>appropriate. However, t requires that the underlying data be normal, or at
>least not too non-normal. My expectations is that most data sets are not
>nearly "normal enough" to make using t appropriate.
>
>So, if we do not know the population standard deviation and we cannot
>assume a normal population, what should we be doing-as opposed to just
>using the CLT as most business statistics books do?
>
>Ronny Richardson
>
>
>Ronny Richardson
>
>
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_________________________________________________________
dennis roberts, educational psychology, penn state university
208 cedar, AC 8148632401, mailto:[EMAIL PROTECTED]
http://roberts.ed.psu.edu/users/droberts/drober~1.htm
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