I thought there was a chance it would hinge on "better". Since it was
"never" that got the emphasis, I thought I'd ask. The problem for me
with the statement "Z is NEVER a better test for the mean under
circumstances they are likely to encounter [in psychology]" is that it
reads like an indictment. While technically correct in some sense,
the use of percentiles of the standard normal distribution in place of
those from the t distribution for large samples doesn't make much
(any?) difference, so the NEVER rule struck me as unnecessary
overkill. In fact, in place of the precise 0.05 two-sided critical
value of 1.96, many people use 2, which is the critical value for a t
with 60 d.f.
> 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.
I would think that for percentiles of the t distribution to be more
right than percentiles of the standard normal distribution for large
degrees of freedom, underlying normality would be critical. However,
I haven't done any formal study of this and will defer to anyone who
has.
> 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.
So maybe z is sometimes better? In fact, it's hard to imagine
circumstances where anyone dealing with real data will not be using a
computer, if only to establish an audit trail. Since software
insists on using t, the question is moot for all practical purposes.
