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st: RE: When is the t-test appropriate?
> I would like to compare the means of two very small random
> samples (n1=3,
> n2=7). Both samples were drawn from populations of unknown
> distribution. Is
> Stata's ttest appropriate in such a situation, or does it
> require the
> underlying populations to be normally distributed? If ttest is not
> appropriate, can anyone suggest are more appropriate method
> for testing the
> difference of the means?
Lots of answers are possible on different levels. Whatever
one says in one sense is likely to be wrong in another,
but here's my take:
1. Supposedly, the t test takes account of your sample
sizes. It is possible that your data are such that the answer
is clearcut even with very small sizes, but perhaps
more likely is that you will fail to get a significant
result (at conventional levels). A common but not universal
interpretation of that would be that it indicates not so much
that there is no difference, but more that the sample sizes
are not large enough to indicate it reliably. Some would
say that this is the main (or even only) role of a significance
2. At the same time, there is a underlying assumption
of normality, whatever the sample size. Nevertheless,
there are many theoretical results, much practical
experience and a lot of folklore to suggest that
moderate non-normality may not be a problem. (Naturally,
what is "moderate"?) In addition, other assumptions
such as equal variances and independence may be much
more critical. For an outstanding discussion with much more
precision, see Rupert G. Miller's "Beyond ANOVA".
3. Calculate confidence intervals and look at them
graphically. There are various possible warning signs, for
example if one confidence limit is an unattainable
value (e.g. physically, biologically, economically
impossible) then you have an enigma wrapped inside
a riddle within a mystery, or whatever it was that
Winston Churchill said on a totally different topic.
5. As Ricardo Ovaldia suggested, you could try a
different approach, such as Mann-Whitney. That,
however, tests a different hypothesis and says
nothing directly about means.
6. Show us the data.
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