Bonferroni correction versus the F-test Many sources emphasize the importance of the F-test p-value in multiple regression and justify this in terms of p-hacking. It's kind of intuitive that if you can't reject the null hypothesis that all coefficients are 0, then it's silly to conclude that any individual coefficient is nonzero. And the false discovery rate for any statistical test is a fine explanation for why you might end up with a very small p-value on an individual coefficient even if the F-test p-value is large.
But there is an alternative story which is that you can avoid p-hacking by applying a Bonferroni correction. This too makes sense.
So what exactly is the relationship between the F-test and the Bonferroni correction? Don't they both kind of solve the same problem? If so, why do we need both?
I'm especially interested in the following edge case: Suppose you observe a regression with, say, the following figures:

*

*5 predictors

*F-test p-value = 0.2

*one predictor's p-value is 0.002.

Now the F-test says 'nothing to see here'. But if we use Bonferroni instead, we end up with that coefficient having a p-value of 0.01, which is quite significant by any conventional standard. So which do you believe?
 A: The Bonferroni correction is a conservative correction that attains its bound when the rejection regions for the individual tests are disjoint (which is hardly ever the case).  Since it is a conservative correction it will tend to overcorrect for the multiple comparison problem.  If there are a large number of comparisons this will lead to substantially lower power, with a higher chance of false negatives.  By contrast, the F-test is neither overly conservative nor anti-conservative.  It forms a test statistic that takes account of the evidence for a non-zero effect operating on one or more of the explanatory variables.
I would generally recommend use of the F-test for multiple comparisons in regression.  The test is designed specifically to test sets of explanatory variables together, so it is perfect for dealing with multiple comparisons.  As to your example, I'm not convinced that your stipulated p-values are even possible, so I don't know that it is a good way to look at the problem.  In any case, given a choice between the two methods I recommend you use the F-test.
