Richard Lehman wrote:
>
> A colleague sent me this note.
>
> >A statistics question.
> >
> >Temperatures taken from different portions of a stream:
> >
> >Portion 1
> >16.9
> >17
> >15.8
> >17.1
> >18.7
> >18
> >
> >mean = 17.25
> >variance = 0.995
> >
> >Portion 2
> >18.3
> >18.5
> >
> >mean = 18.4
> >variance = 0.02
> >
> >Do these portions have different temperatures?
> >
> >Obviously the variances are unequal and a 2-sample [unequal variance]
>
> t = 2.74 w/ 5 df p = 0.037.
>
> >No problem.
>
> >But (and this is where I am perplexed), a pooled [equal variance]
>
> t = 1.54 w/ 6 df p = 0.17.
>
> >Why is the less conservative pooled t giving a lower t-value? Are the
> >variances so uequal (and the one so close to zero) that the formula is
> >messed up?
Because hypothesizing equal variance requires us to discount the
apparent evidence that Population 2 varies very little, and to conclude
that if we took lots of samples from Population 2, those sample means
would vary rather widely. In fact, X-bar-two would be the primary source
of variation in (X-bar-two - X-bar-one), due to the smaller sample size;
and the observed discrepancy between it and X-bar-one would not be very
surprising.
Which is right? It is certainly _not_ "obvious" that the variances are
unequal, as in fact the F distribution tells us that we would expect to
see such a difference in variance, in that direction alone, about one
time in 10, were they equal. A two-sided p-value of 20% is perhaps cause
for some suspicion of inequality, but hardly an obviosity. Even a
moderately strong prior belief in equality should not be much swayed by
this. On the other hand, in the absence of any _a_priori_ reason to
suppose so, we should not conclude that they _are_ equal.
On the third hand, if we had an _a_priori_ reason to suppose even that
the variances were moderately similar in size, we would have reason to
suspect that the small difference in the second sample was in fact a
fluke - given that such flukes happen, in one direction or another, 20%
of the time!
The real moral: data sets of size two are just not big enough to do
anything with, and the proper solution is to get out there with that
thermometer again and do it right.
-Robert Dawson
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