# Power of a statistical test against a compound alternative hypthosis

I was recently posed a homework question. It will be a few days before I can get in-person help from faculty, so I wanted to ask here. The question was:

A medical resident doing research wants to know if the mean heart rate after a particular type of trauma differs from the healthy population rate of 72 beats/min. They consider a mean difference of 6 beats/min to be clinically meaningful. They also choose 9.1 beats/min as the variation based on a previously published study. How many patients will be needed to carry out the study at 0.0005% significance and 82% power?

I've been struggling to find an answer for the "power" portion of the question, because every reference (between our two textbooks, internet searches, and CV) has had power calculations relative to a specific alternative hypothesis; and as far as I've seen, this is how power is defined.

I've since handed in the homework, and have emailed the professor and a TA. The professor directed me to some lecture notes that supposedly lead to a formula for sample size based on a specified significance, power, and rejection criteria:

$$n = \left[\frac{\sigma(z_\alpha + z_\beta)}{u_1-u_0}\right]^2$$

with $$z_\alpha = \frac{c - \mu_0}{\sigma / \sqrt{n}},\quad z_\beta = \frac{c-\mu_1}{\sigma/\sqrt{n}}$$

and $$c$$ is (presumably, the notes are not very clear) the cutoff value for the critical region.

But as far as I can tell, these still rely on a simple (not compound) alternative hypothesis $$\mu_1$$.

The TA's response was:

There’s actually a alternative hypothesis. There’s maybe some wording problem with this question, it actually means the null hypothesis is that a mean difference of 0 is clinically significant. So the alternative hypothesis would be a mean difference of 6 is clinically significant. Then you can use the formula.

So perhaps I'm just misreading the problem; but I still can't make sense of what the "mean difference of 6" is supposed to be, if not a compound alternative hypothesis (i.e. that $$H_1: Y \leq 66 \text{ or } Y \geq 78$$).

The questions I'd like answered are thus:

1. Is this question bunk?
2. If not, what is the alternative hypothesis? Should I be using the difference of two means (i.e., a paired t test or similar) rather than a test about one mean?
3. Is it true that the power of a statistical test is always with respect to a particular (i.e., simple; non-compound) alternative hypothesis?
• Please add the self-study tag. – Michael R. Chernick Dec 2 '19 at 1:58

With a two-tailed test, a "mean difference of 6" merely indicates that $$|\mu-\mu_0|=6$$. You don't need to worry about whether the difference is positive or negative; focus on using a test statistic of $$|z|$$. Whether that strictly counts as compound or not depends on your point of view.
• Got it: so (presumably) the questions was asking for the power at 6; not saying that $\pm$ 6 was the rejection criteria. I think I know where to go from here... – Peter Dragos Dec 2 '19 at 15:50
• Correct. Power is "if the difference was 6, what's the chance you reject?" ... in this case you then want to look at "How big does $n$ need to be to make that chance 82%?" – Glen_b -Reinstate Monica Dec 2 '19 at 22:47