Power of a hypothesis test My question may be pedantic, but I find the syntax used to describe the "power" of a hypothesis test really annoying, and I just want to either correct my understanding of the concept, or understand why it is, seemingly, mislabeled.
My understanding of "power" is the probability that a test will reject $H_0$, when $H_a$ is true. Just from this wording, the power of a test seems to be the power in favor of $H_a$, not the power against it. In other words, if $H_a$ is true, you want your test to reject $H_a$. However, my text, Introduction to the Practice of Statistics, used as a text book at our community college, states that this probability is "the power against $H_a$". 
Why do they call the probability that $H_0$ is rejected when $H_a$ is true the power against $H_0$ instead of the power in favor of $H_a$, or simply the power of $H_a$?
 A: I have not checked out your text, but your understanding is correct. 
The alternative hypothesis (Ha) is usually stated vaguely as something like the difference between the two population means is not zero. But for the purpose of computing and interpreting power you need a definitive Ha, say that the difference between population means equals 10 (or some value). If that Ha is true, and if you accept all the assumptions of the test, power is the probability that random sampling of data from the two populations with the specified sample size will result in a P value less than alpha. 
So yes, it is the power against the null hypothesis and for the alternative. 
A: It seems to me that it is the power of the test itself, rather than the power against either of the hypothesis.  If there is very little data, the test will be unable to reject $H_0$ whether it is false or not, so the test itself has little power to reveal anything about the problem.  If we have lots of data, we will be able to confidently expect to reject a false $H_0$ even if the likely effect size under $H_a$ is rather small, so the test is powerful.
Perhaps we should think of the power of a test being somewhat analogous to the (optical) power of a microscope or telescope, in that it gives an idea of how fine a distinction we can reasonably expect to resolve.
I suppose you could however argue that if we are unable to reject the null hypothesis using a test with high power, then this is "powerful" support for the idea that $H_a$ is probably false (as if $H_a$ were true, the test would be highly unlikely to fail to reject $H_0$).
See statistical power for info on the statistical meaning of "power".
