I share Paul's concern about fitting higher order polynomials. My
impression was that Don Burrill was not advocating this and prefered
the exponential model Paul mentioned, presenting the polynomial
approach as one used by some students. I think Don was trying to
indicate the best possible solution GIVEN one had decided to fit
polynomials. And for Murray's data, a change in conditions certainly
suggests a piecewise model. Paul rightly emphasizes the importance of
using all the available information, but I see Murray's example as an
explanation of WHY a cubic gives a close fit empirically, even though
it may not be a good model of the underlying structure.
It is often the case that we don't have much understanding of
why a relationship takes the form it does, nor what the true
functional relationship might be. For the girls' heights, I doubt
very much that the true relationship is cubic -- and it certainly is
NOT cubic if we extend the range of ages. However, a cubic DOES
summarize what I take to be the important facets of the data given
within the range of ages considered, i.e., rapid growth followed by
slower growth followed by another period of rapid growth.
For teaching, the main message is that higher order polynomials are
rarely good models of the underlying structure, but may occasionally
provide reasonable summaries of the data.
Tom Moore's original request for data well fit by a cubic tacitly
implied that it's actually pretty unusual for a cubic (or higher order
polynomial) to be a good choice. Perhaps Tom even doubted that it
would EVER be a good choice, and wondered if anyone could provide a
counterexample!-)
_
| | Robert W. Hayden
| | Work: Department of Mathematics
/ | Plymouth State College MSC#29
| | Plymouth, New Hampshire 03264 USA
| * | fax (603) 535-2943
/ | Home: 82 River Street (use this in the summer)
| ) Ashland, NH 03217
L_____/ (603) 968-9914 (use this year-round)
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