Alan McLean <[EMAIL PROTECTED]> wrote in
sci.stat.edu:
>The use of the t distribution in inference on the mean is on the whole
>straightforward; my question relates to the theory underlying this use.
>If Z = (X - mu)/sigma is ~ N(0, 1), then is T = (X - mu)/s (where s is
>the sample SD based on a simple random sample of size n) ~ t(n-1)?
Yup.
This is one example of the general principle: In doing a hypothesis
test, you compute the value of a suitable test statistic, which has
a known distribution. Then you compute the probability of getting
that test statistic (or one even more extreme) for a sample the size
of yours, just by random selection. That probability is your p-
value, which you compare to alpha. (The classical approach instead
computes the critical value of the test statistic based on alpha,
and compares the actual value of the test statistic to the critical
value.)
>So my question is: how do YOU explain to students what a confidence
>interval REALLY is?
I explain it this way:
It is the result of a process. For example, a 95% confidence
interval is the result of a process that, for 95% of possible good
random samples, gives you an interval that contains the true
population parameter (e.g. mean). So if you take a good random
sample and compute a 95% confidence interval, there is a 95% chance
that the true population parameter is within the computed interval.
The problem, of course, is that you have no way to know, for any
given sample, whether this is one of the 19 out of 20 that will lead
to a correct confidence interval or one of the unlucky 1 out of 20.
I like Eliot Cramer's "bet" idea, and I think I'll steal it. :-)
--
Stan Brown, Oak Road Systems, Cortland County, New York, USA
http://OakRoadSystems.com/
"My theory was a perfectly good one. The facts were misleading."
-- /The Lady Vanishes/ (1938)
.
.
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