This paradox is a great example of why I'm uncomfortable with the idea of coming up with a p-value when you're comparing a sample mean to a point value in the first place.

If your null hypothesis amounts to little more than a scalar value that exists in a vacuum, you haven't really got a null hypothesis. If your null hypothesis is that your results will match what was seen in some other data set collected by some other person who may or may not have been using the same protocol as you, your null hypothesis describes a parallel universe and there's no way to draw an apples-to-apples comparison to it and your alternative hypothesis, which concerns data that did not come from that parallel universe.

So I guess my answer to all questions would be, "Mu."