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 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
