Robert J. MacG. Dawson wrote on 11/3/00 2:16 PM:
>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.
To agree with Dr. Dawson,
I checked with three methods of testing equality of variance and found
none were signficant:
Test F DFs P
Brown-Forsythe 1.2690 1 6 0.3030
Levene 2.0648 1 6 0.2008
Bartlett 2.1103 1 € 0.1463
But, I'd be uncomfortable with data with only 2 observations in one
group. If you are stuck with the data, then you are stuck with the equal
variances t-test because there is insufficient evidence on which to base
a decision to use the unequal variance t-test.
Paul Bernhardt, M.S.
University of Utah
Dept. of Ed Psychology
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