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Re: What is the Logic Behind Hypothesis Testing?
Posted:
Nov 3, 2006 11:47 AM
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Central limit theorem dictates that the sum of the i.i.d random variables
would approaches normal distribution.
"Joel Daniels" <jdaniel4@uiuc.edu> wrote in message
news:1162362555.542791.116130@i42g2000cwa.googlegroups.com...
> 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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