An intro stats student wrote (concerning the 95% CI for the mean):

>All we are trying to do is establish the range over which we are 95%
>certain our sample mean will reside in, if the alternative hypothesis is
>true. Basically we find the left and right critical values of our
>alternate hypothesis, and that is our range. 

And I replied:

Hold on a second.  You KNOW what your sample mean is.  You don't need a
range over which you are 95% certain it falls in.

To calculate a 95% CI for the POPULATION mean, you take your sample mean
plus and minus the Standard Error times the critical value of z (or t,
depending on whether or not the population variance is known), where the
critical value is with alpha = .05 2-tailed.

Let LL = lower limit of 95% CI
    UL = upper limit of 95% CI
    mu = population mean
    X-bar = sample mean
    SE = standard error of the mean
    CV = critical value of z (or t if Population SE is not known) with alpha = .05 2-tailed

LL = X-bar - SE * CV
UL = X-bar + SE * CV

You may then see statements like this:

	p(LL < or = mu < or = UL) = .95

That is, the probability that the population mean falls between the Lower
and Upper limits of the 95% CI = .95.  But a word of caution about this. 
I'll quote from David Howell's textbook, Statistical Methods for
Psychology (4th ed, p. 204):

----------- Start of quote from Howell -------------------------------

Statements of the form p(1.219 < mu < 1.707) are not interpreted in the
usual way.  The parameter mu is not a variable--it does not jump around
from experiment to experiment.  Rather mu is a constant, and the interval
is what varies from experiment to experiment.  Thus, we can think of the
parameter as a stake and the experimenter, in computing confidence limits,
as tossing rings at it.  Nintey-five percent of the time, a ring of
specified width will encircle the parameter; 5% of the time, it will miss.
 A confidence statement is a statement of the probability that the ring
has been on target; it is not a statement of the probability that the
target (parameter) landed in the ring.

----------- End of quote from Howell ------------------------------------


Cheers,
Bruce