F-test in Regression and Crossvalidation I am wondering what is the better measure of predictive power in a multivariate regression setting.   
So an F-test for regression tests "the utility of the model", or exactly, if any of the $\beta$ coefficients are non-zero.  When the null is rejected, the model is considered "useful"; so, useful for prediction?  Crossvalidation also can help tell you if your model is useful for prediction.  Can you talk about them using the same phrasing?
What about the $R^{2}$ statistic?  Say $n$ is large, and the degrees of freedom in your model is small.  It is pretty unlikely overfitting has occurred.  What can you say about the predictive power of your model in this case when $R^{2}$ is high? medium? zero?
Examining the predictive power of some model is a huge subject, but do F-tests play into that?  Has crossvalidation become so popular because of machine learning methods where statistical tests don't exist?  Was it popularized to fill that void in algorithmic models, or do they really address different issues? 
 A: 
So an F-test for regression tests "the utility of the model", 

Well, no, it really doesn't. A rejection doesn't imply the model is actually useful at all.

or some such carefully worded statement.

If you mean that 'there's a carefully worded statement that applies to the F-test', this is a trivially true, but essentially useless statement.

Crossvalidation basically does the same thing. 

No, it doesn't do the same thing as the F-test, not even approximately.

What is the best way to express what an F-test tells you about the predictive power of your model?

Perhaps something along the lines of "Unless you have an unusual definition of predictive power, the F-test may not directly tell you much about it."
Specifically, the summary of the result of the test would be "$H_0$ was rejected" or "$H_0$ was not rejected".
But let's take a step back and consider something slightly more informative, like the $p-value$. That doesn't really tell you about predictive value (which to me wouldn't tend to change much with $n$, once the sample size was large enough to have reasonable parameter estimates, utterly unlike the p-value.).
If you define predictive power as say $R^2$ (which to me would seem bizarre), then the $F$ statistic is related to it (for fixed model and sample size, at least), but we're now getting some distance from the test and still not all that close to predictive power yet. And if we start doing variable selection or something, $R^2$ is getting further and further from carrying any information about actual predictive power.

Would it be weird to see an analysis of data where someone used crossvalidation and quoted the p-value for the F-test to asses its predictive ability?

Well, weird because the F-test doesn't really address that question.
It seems to me a better question for you might be "What does an F-test actually do?" and a similar question about cross-validation, and maybe resolve exactly what 'predictive power' might usefully mean.
Once you define more carefully what particular thing you're doing cross-validation on, and exactly what the circumstances and assumptions are, it would be possible to make more useful/interesting statements.
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You might find this a good starting point for basic discussion of cross-validation from a statistical point of view:
http://robjhyndman.com/hyndsight/crossvalidation/
You might notice there's hardly a mention of hypothesis tests at all, except a couple of sentences of precaution about circumstances in which they don't make much sense.
