Gary McClelland, Dick Startz, and A. G. McDowell all have pointed to the error that I was making in my argument that the sampling distribution of b should be skewed. I have taken their point and played around a bit more, and have finally resolved the issue to my satisfaction at least.
The sampling distribution of b is, in fact, normal, just as the theory predicted. But the sampling distribution of beta is not normal. My argument had been that beta is a simple linear transformation of b. But in fact, over repeated samples, the "constant" that multiplies b to get beta (names s(i)/s(0) is not a constant, but a random variable. Both s(i), the standard deviation of the predictor and s(0), the standard deviation of the criterion, are random variables. On top of that, they aren't even normally distributed.
I put together a pdf document that illustrates the samping distributions of b and beta. I'd be happy to send it to anyone who would like it. Or you can download it at http://www.uvm.edu/~dhowell/StatPages/More_Stuff/SampDistRegCoeff.pdf.
Thanks for your help.
Dave Howell
David C. Howell
Professor Emeritus
University of Vermont
New address:Professor Emeritus
University of Vermont
David C. Howell
3007 Barton Point Circle
Austin, Tx 78733
http://www.uvm.edu/~dhowell/StatPages/StatHomePage.html
http://www.uvm.edu/~dhowell/gradstat/index.html
