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  • $\begingroup$ Thanks Macro. Very helpful answer. I'm glad you included the part about what, exactly, the p-value is testing. It makes a lot of sense that the p-value would be so low considering how close to 1 the slope is. It seems to me, in light of your answer and @jedfrancis', the r^2 value describes that 'cloud' of data points around the line of regression. Excellent! That's much more clear now! $\endgroup$ Commented Jul 19, 2011 at 19:56
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    $\begingroup$ @Macro (+1), fine answer. But how does the "strength of the relationship" depend on the "size of the intercept"? AFAIK the intercept says nothing at all about correlation or "strength" of a linear relationship. $\endgroup$ Commented Jul 19, 2011 at 21:32
  • $\begingroup$ @whuber, you're right - the intercept is irrelevant and definitely doesn't change the correlation - I was thinking about the regression function $y = 10000 + x$ vs. $y = x$ and thinking somehow of the second one being a stronger relationship (all else held equal), since a greater amount of the magnitude of $y$ was due to $x$ in the latter case. Doesn't make much sense now that I think about it. I've edited the post. $\endgroup$ Commented Jul 19, 2011 at 21:36
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    $\begingroup$ @macro Excellent answer, but I would stress (for those new to this subject) that R^2 can be very low even with a strong relationship, if the relationship is nonlinear, and particularly if it is nonmonotonic. My favorite example of this is the relationship between stress and exam score; very low stress and very high stress tend to be worse than moderate stress. $\endgroup$ Commented Jul 20, 2011 at 10:19
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    $\begingroup$ @macro Yeah, your answer was good, but I have worked with people who don't know a lot of statistics, and I've seen what happens ... sometimes what we say is not what they hear! $\endgroup$ Commented Jul 21, 2011 at 10:42