On Wed, 2 Aug 2000, chris (who, from his e-mail address, clearly does not
desire a personal response) wrote:
> I have a doubt as to what is the assumption for normality in a t-test.
>
> Two possibilities:
These may perhaps not exhaust the universe of discourse.
> I think it is for the sample data I have,
Sample data are discrete and the sample is of finite size.
The normal distribution is continuous and unbounded (any value between
minus infinity and plus infinity is legitimate); if applied to a
distribution of discrete elements, one would need an infinitude of such
elements.
It follows that no sample can be normally distributed; although its
empirical distribution may be more or less well approximated by a normal
distribution.
> while a dear friend says it relates to the population and
> Central Limit Theorem.
The assumption applies to the population from which the sample is
[assumed to be] randomly drawn. The "normality assumption" does not
require the CLT; if the population is normal in the first place, then
the distribution of the test statistic ( t ) will follow "Student"'s
t-distribution. If one is unwilling to assume the underlying population
to be normal, one may be willing to invoke the CLT in order that the
sampling distribution of the _mean(s)_ to be examined be (at least
approximately) normal, so that the distribution of t will behave.
> "I went back to the reference on the t-test normality assumptions and
> the assumption refers to the individual values, not the averages.
A half-truth. In the text you cite, the author chose to start with a
normal random variable, probably to avoid having to argue about the
sampling distribution of the sample mean. If the random variable is
normal, the sampling distribution of sample means will be normal.
(After all, a value of t is only a standard score: the deviation of a
sample value from a population mean, expressed in units of the standard
deviation of the variable. For that value to have "Student"'s t
distribution for its sampling distribution, it suffices for the deviation
to be normally distributed with mean zero, and the standard deviation (in
the denominator) to be distributed as chi. These conditions are met _a
fortiori_ if the underlying random variable is normal, and may be
adequately approximated if the underlying variable is not normal so long
as an appropriate CLT applies to the sample mean and to the sample
standard deviation.
> In part [the text] reads:
>
> "Suppose that x is a normal random variable with unknown mean m and
> unknown variance s2. We wish to test the hypothesis that the mean
> equals standard value mo. Because the variance is unknown, we must
> make the additional assumption that the random variable is normally
> distributed.
For "must" read "find it convenient".
> The normality assumption is needed to formally develop the
> statistical test,
Rather, The normality assumption permits us readily to prove a theorem
that specifies the distribution of the test statistic.
> but moderate departures from normality will not seriously affect the
> results."
True.
< snip, the rest >
-- DFB.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
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