>
> 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)?
The answer is of course, yes.
>
> The expression P(Xbar - 1.96 x SE < mu < Xbar + 1.96 x SE) = 0.95 is a
> perfectly good prediction interval - it expresses the probability of
> getting a sample mean which satisfies this inequality.
Not quite unless you intend SE to be a population parameter.
Lets assume that is what you meant.
>
> Now replace the RV Xbar by the observed sample value to give the
> interval: xbar - 1.96 x SE < mu < xbar + 1.96 x SE. This is of course
> the confidence interval on the population mean mu.
>
> Whatever is said in the text books, this is understood by most people as
> a statement that "mu lies in the interval with probability 0.95" - or
> something very close to this. In effect, we define a secondary notional
> variable Y which imagines that we could find out the 'true' value of mu;
> Y = 1 if this true value is in the confidence interval, = 0 otherwise -
> and we estimate the probability that Y = 1 as 0.95.
>
> I have been teaching statistics for 30-odd years and have become more
> and more disillusioned with the treatment of confidence intervals in the
> text books!
>
> So my question is: how do YOU explain to students what a confidence
> interval REALLY is?
It is really an animal created by a committee -- Neyman and
Pearson*.
For most common uses, including the example above, which
involve pivotal quantities, one can justify the naive view
about such intervals, either with a fiducial or likelihood
argument. See Hacking's Logic of statistical inference for
details. In point of fact, most of us realize, if we are
honest, that a confidence interval can be interpreted only
by adopting the naive view. Several noted authors have
pointed this out, Hacking among them.
One way to help the students is to tell them the truth about
it, and make them comfortable with the way they will find
themselves interpreting these things.
*Not quite true since it was really Neyman alone, but it has
all the aspects of a committee product.
>
> Regards,
> Alan
>
> --
> Alan McLean
> [EMAIL PROTECTED]
> +61 03 9803 0362
--
Bob Wheeler --- (Reply to: [EMAIL PROTECTED])
ECHIP, Inc.
.
.
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