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Topic: pooled t test vs "unpooled" t test
Replies: 1   Last Post: Nov 6, 1996 10:34 AM

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Posts: 144
Registered: 12/6/04
pooled t test vs "unpooled" t test
Posted: Nov 4, 1996 8:39 PM
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To this observation

>> -- My guess would be that if a stat-program presented just one
>> "two-sample t-test", it *would* be the "Student's t-test" which pools
>> the variances. (The t in "t-test" is not capitalized.)

Bob Hayden commented

> Minitab only pools the variances if you explicitly ask it to. I
> think Minitab's default is the correct default. You lose very little
> if the variances ARE the same and you gain a lot of they are not.

In "What is Statistics?" (Chapter 1 in the MAA's Notes Number 21,
_Perspectives on Contemporary Statistics_), David Moore says:

"The pooled-sample t test ... is somewhat robust against unequal sigmas
if the sample sizes are equal, but not otherwise." He uses T_p for the
pooled statistic. Then, arguing for the "unpooled" statistic, labeled T,
he asserts that "A substantial literature ... demonstrates the accuracy
of this [i.e., the "unpooled" T] approximation for even quite small
samples, and demonstrates in addition that when in fact sigma_1 =
sigma_2, using T sacrifices very little power relative to T_p." (p.15)

It's this kind of "expert testimony" that is missing from so many
textbooks at the elementary level--the kind of testimony that would help
us make wiser decisions about the relative merits of various procedures,
when working with students.

(This debate about the two t tests is not all one-sided, of course. I
believe I recall Paul Velleman arguing that a good reason to teach the
pooled test is that it generalizes readily to ANOVA.)

Here's another comment by David Moore whose essence is missing from most
elementary textbooks: "... the F ratio for comparing variances is almost
worthless. ... The reason for this ... is that no data are exactly
normal. ... the F ratio is so sensitive to even small departures from
normality as to be almost useless." (same source, p. 14)

Bruce King
Department of Mathematics and Computer Science
Western Connecticut State University
181 White Street
Danbury, CT 06810

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