[I suspect this reply will not be broadcast to ANZstat, as I am not a member of that list; you may (or not!) want to forward it, Alan.]
On Mon, 18 Nov 2002, Alan McLean wrote: > I have a couple of questions, one of which has been bubbling round > in my mind for some years, the other is more recent. The recent one > is the following: > > 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)? Short answer: Yes. Longer answer: the number of degrees of freedom for the t distribution for such a statistic is the number of degrees of freedom associated with the variance estimate (well, with its square root) in the denominator. > My second question is on the matter of confidence intervals. <snip> > > 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. > > 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. Minor quibble: Provided "SE" is the population standard error of the mean (and supposing that you're specifying a 95% C.I.), which is consistent with the notation you specified. > 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. Interesting concept, I'll have to think about that "notional variable" a bit. > 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! I think the earlier approaches that began with hypothesis testing before introducing the idea of confidence intervals were superior to what I've been encountering of late, where C.I.s are introduced first (often, one suspects, before many students have managed to internalize the idea of probability at all thoroughly), and hypothesis testing appears later. > So my question is: how do YOU explain to students what a confidence > interval REALLY is? A C.I. is an observed instance of a random variable, representing the range of values that one might specify in a null hypothesis and NOT have the hypothesis rejected. Where an acceptance region (which I hope I'll have had occasion to explain previously!) is an interval centered on the value specified in a null hypothesis, and represents the range of possible values of the sample mean that would NOT lead to rejection of that hypothesis; a C.I. is an interval centered on the observed sample mean (which is why it is a value of a random variable: tomorrow, if you went out and looked again, you'd probably find a different value of the sample mean, hence a different C.I.), and represents (as above) the range of values of possible null hypotheses that could NOT be rejected under the conditions of the current experiment. I usually added, in introducing a C.I. in the first place, that sometimes one has an obvious null hypothesis (that is, an obvious null- hypothetical value of a parameter) to test, and in that circumstance an hypothesis test is clearly appropriate. But sometimes there isn't an obvious value to specify for mu (or sigma, or rho, or beta, or ...), and then one might be interested in knowing what (potential) values of mu (or whatever) would be consistent with the data in hand. Is this any help? -- Don. ----------------------------------------------------------------------- Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 [was: 184 Nashua Road, Bedford, NH 03110 (603) 471-7128] . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
