Re: Regression significance conundrum

I'm going to reply to myself rather than reply individually to the
several excellent replies my question has generated (I haven't been
ignoring the answers, but my news server has been taking some sick
leave for the last few days). I've been reassured that the answer to
my question was not obvious by the way every contribution has added
new information, and no-one has taken issue with anyone else's
contribution.

I've drawn a number of valuable conclusions about what I should do
differently next time.

Some matters arising (not individually attributed):

The point about correct interpretation of the p values in this
context, namely that they are "conditional" - that is, they measure
the additional significance of each term given the other two - is
clearly the most important issue here.

The suggestion to standardize the data is a good one. I had been
worrying about the collinearity issue, but it did not occur to me that
standardizing the data would remove it. Doh.

I did.start with a good old scatterplot; it was on the basis of the
original scatterplot that I decided there might be some slight
curvature. The suggestion was made that I should have made that
decision on the basis of examining the residual plots having fitted
the linear model,; presumably that is only if the required decision is
linear vs non-linear? I mean, if there is clear curvature, then
presumably I fit e.g. a quadratic model, and examine the residual
plots in order to decide whether or not higher-order terms might be
warranted?
.