In article <[EMAIL PROTECTED]>,
Ronny Richardson <[EMAIL PROTECTED]> 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.
Wrong! It states, in part, that the distribution of the
sample mean gets closer (in the sense of convergence in
distribution) to the normal distribution as the sample
size increases. The distribution is never normal unless
the original distribution is normal.
>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 not overlooking anything. The rate of convergence
is usually not the greatest.
>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.
The approximation of the distribution of the t statistic
to the t distribution is similar of that of the mean to
the normal. Symmetric tests are somewhat better for most,
and the t-test is usually used two-sided.
>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?
1. Specify your real problem; what are the consequences
of taking wrong actions, and how important are they?
2. If the results are not too sensitive to the cut
point, and your problem is similar to those for which
the t-statistic or the normal distribution is usually
used, go ahead and use it.
3. If not, consult a good mathematical statistician.
It is even possible that new recipes need to be used.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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