What is the Logic Behind Hypothesis Testing?

Greetings All,

I am taking an introductory statistics course and I am struggling with
the concept of hypothesis testing. I understand all the steps and can
blindly perform the task, but I do not understand what exactly the
result means and why anybody would care to find it.

I have a myriad of questions, but I think I found one concrete way to
address my confusion:

I'll start with a textbook-style problem:

A 1-value sample "x" is taken a random variable "z" that follows a
normal distribution with standard deviation 1, but unknown expectation
value (mean).

Test (with a 90% confidence level) the null hypothesis
H_0: mean of z = 0
against the alternative hypothesis that
H_A: mean of z <> 0

To solve this I need to define a "critical zone," such that the
probability that a random sample of z falls in this critical zone is
10%. For some reason that I do not fully understand, I can further
stipulate that the "appropriate" critical zone is divided equally
between the two extremes.
So the critical zone is roughly

(-infinity,-2) U (2, infinity)

Next, I check to see if my random sample x falls within the critical
zone. If it does then I reject the null hypothesis, otherwise I fail
to reject the null hypothesis.

Great, but why this choice of critical zone? I understand why 10% of
the area under the p.d.f of "z" must lie in the critical zone, but I do
not understand why this zone should be equally distributed between both
extremes. I know that this "rule" depends on the distribution. For
example imagine that I repeated the problem above, but with a p.d.f for
z that looks like a capital "M" with each vertical bar of the "M" 1
unit away from the mean. Formally this distribution is:

For z < mean - 1 ....... p.d.f(z)=0
For mean -1 <= z <= mean + 1 ....... p.d.f(z)=abs(z-mean)
For mean + 1 < z ..... p.d.f(z)=0

In this case, I know that my sample value "x" CANNOT be 0, if the mean
of z is 0. In other words, if I see a sample value of exactly 0, then
I know 100% for sure that my null hypothesis is false. Logically, this
implies that I should be suspicious if I see a value of 0.00012698...
Yet if my critical zone is the area near the extremes of my function,
then I cannot reject my null hypothesis on a sample value of
0.00012698, even at the 10% level!

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? If so, can anyone help me prove it,
or even write a simulation that shows its validity? Every time I try,
I get stuck because I don't have any idea what a hypothesis test really
says in the first place, so I can't prove that methodology X gives my
value Y, when I don't know what the definition of Y is.

Any help would be greatly appreciated.

Thanks,
Joel Daniels

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