> Robert Dawson wrote:
> > Exactly... An example - we've been using Devore & Peck, which
> > unfortunately introduces the Z test for the mean, supposedly for
pedagogical
> > reasons but without nearly a strong enough indication of this. A lot of
> > students infer a rule "if n>30 use z rather than t" despite my repeated
> > statements that Z is NEVER a better test for the mean under
circumstances
> > they are likely to encounter [in psychology]. Of course, if they are
cutting
> > lectures that day they won't hear the warning...
>
> Okay, I'll bite. Why?
Recall that "the" z test for the mean is actually two often-confused
tests. The first, the "Z test with sigma", involves exact prior knowledge
of sigma. This is an artificial situation very unlikely to arise if the
variation is intrinsic to what is being measured. If the variation comes
from a separate source (eg, instrumentation) it is possible that one might
know sigma but not mu - but this is more of an engineering scenario, and
probably oversimplified even then.
The "Z test with s" is nothing but an unnecessary approximation of the
t distribution for n>>1 degrees of freedom by the z distribution. The most
that can be said for it is that if n is large it is not wrong by very much.
However:
-inasmuch as the outcomes, p values, or confidence intervals obtained
differ from those of the t procedures, the z outcomes are wrong and the t
procedures are right. Z is never mathematically better.
-Students need to learn how to use both tables anyhow. Using "z above
thirty" does not reduce the amount students need to learn. If they are
using a stats package the same principle applies. For this and the next two
reasons, z is never pedagogically better.
Caveat: Old fashioned t tables fashioned after the tradition the Church
of the Holy 5% make it hard to compute p values that are not round numbers.
See Devore & Peck's 3rd edition, or my article "Turning the Tables - a
t-table for Today" in JSE
a couple years ago, for alternatives.
-The test-selection decision process is made more complicated if the "z
over thirty" rule is added, not less so.
-Students tend to somehow twist the "z over thirty" rule around to say
(to them) "t is incorrect over thirty". This could be embarrassing to them
in later life (eg, if they were refereeing a paper and demanded that the
author change a t test to a z test).
-Robert Dawson
