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The statistics question asks which would be the best method to determine if your data would best be fit with a linear regression model or a quadratic model.

I would think that the residual plot would be the best method. The other 2 choices are the shape of the scatter plot and the correlation coefficient.

I have seen scatter plots that look very liner and have a $.9$ R value, yet when you look at the residual plot you can see a distinctive pattern or curve.

So, isn't the residual plot not only the best but the ONLY method? I guess I'm no longer sure what the R value really does, since R values of .9 can still NOT be a good linear fit.

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    Is there much value in looking at the r value? – 2017-02-15

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I would go with a q-q plot if given the opportunity, but residual plot is best from choices you are given.

The $R^2$ captures the basic notion of goodness of fit, but it is insufficient here. For example, if your linear regression leaves residuals which are not looking like Gaussian errors, even with a large $R^2$, you can likely do better as there seems to be some inherent effect in the data not yet described by your model...