Re: What is the Logic Behind Hypothesis Testing?

I _think_ that the truth is that you want the critical zone to be some
zone such that for all "a" within the zone, and for all "b" outside if
it this inequality holds true:

p.d.f (a) <= p.d.f(b)

This leaves me stuck if I have a uniform distribution... but I am OK
with that, because I can't envision how a standard hypothesis on a
uniform distribution could tell you anything interesting anyway.

I came up with this "rule" for the critical zone when the alternative
hypothesis is

H_A: mean of z <> 0

based on intuition. Is it right?

Generally speaking, your intuition is correct. In hypothesis testing, your rejection region (what you call your "critical zone") should consist of those values which you would be most "surprised" to see if your null hypothesis were true. If you restrict yourself to two-sided tests on unimodal symmetric distributions (things with only one "bump"), then your rejection region will, I think, always be as you describe. But imagine a distribution that looks like a Normal but has one small "bump" in either tail. Then your rejection region may still be (-inf, -2) U (2, +inf), but you could fix things such that f(-3) > f(-2) and f(3) > f(2).

Also, a couple of thoughts about your M-shaped distribution example: Don't forget that you are testing the hypothesis that the *mean* of the distribution is zero. In this case, you'd still want to reject for large values of x (or, more usually, the sample mean). Your statement about wanting to reject if you observe values of x near zero would be reasonable if your null hypothesis were something like "the data came from an M-shaped distribution with mean zero" (depending on what your alternative hypothesis was).

-J
.

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