One thing is clear from the snow storm of postings - A dog house is not a house dog.
Not being facetious here; it's just that my students, and I as well, can get the meanings of the terms confused. Between an estimated standard deviation and a standard deviation... Between an average and a mean..... Between a confidence interval and a prediction interval... Between a probability and chance likelihood... I plan to write out all the postings and see if I can run them out in a straight logic line. Maybe if I'm careful enough with the definitions/meanings, it will become more clear. the truth is that statistical analysis gets us home more often than not. Whether with 95% confidence, or 93%, or 97%, remains an issue. Cheers, Jay Alan McLean wrote: > Hi to all, > > 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)? > > My second question is on the matter of confidence intervals. In my > explanation here I am using the convention of upper case letter to > represent the random variable, lower case to represent a value of the > variable. > > 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. > > 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? > > Regards, > Alan > > -- > Alan McLean > [EMAIL PROTECTED] > +61 03 9803 0362 -- Jay Warner Principal Scientist Warner Consulting, Inc. 4444 North Green Bay Road Racine, WI 53404-1216 USA Ph: (262) 634-9100 FAX: (262) 681-1133 email: [EMAIL PROTECTED] web: http://www.a2q.com The A2Q Method (tm) -- What do you want to improve today? . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
