Hypothesis testing on proportions if $H_0: p = 0$ A new medical procedure is tested on 225 subjects ; the observed success rate is 8%. The standard test is used in these cases is to test the possibility that the real success rate is actually 0%.
I understand that the sampling distribution of proportions is a normal distribution centered at $p$ and with standard deviation 
$\sqrt{pq / n}$. How does one perform a hypothesis test on a proportion then if one wants to test the hypothesis $p=0$?
 A: In short: as @dsaxton commented, if you observe any success, reject p=0; otherwise don't reject.
Testing $H_0: p = 0$ (or $H_0: p = 1$) is quite different from testing any other value of p, because the null hypothesis $p=0$ means that no success will be produced, never. Therefore, if we observe one success (even if it's just one in billions and billions), we can be absolutely sure that $p$ is different from 0.
When testing any other value of $p$ (let's say $H_0: p = 0.5$) and the null hypothesis is true, different samples can produce different outcomes, usually with a sample proportion near 0.5 and we can discuss how far from 0.5 need it to be to reject that $p$ equals 0.5. However, if the null hypothesis is $H_0: p = 0$ and it's true, all samples will have the same proportion (no successes) and any other proportion sample proportion is impossible under the null hypothesis. Therefore, if we get any other proportion we must reject the null hypothesis.
As a side note, here the usual approximation of binomial with a normal doesn't work, because one of its conditions is $n·p>5$, and here $n·p=0$. Anyway, it's easy to use the binomial, here - in fact so easy that I wouldn't call it random distribution, because it's just $P(0)=1$ and $P(\text{anything different than 0})=0$. With this distribution and using test language, if you don't get any success p-value=1 and if you get one or more successes p-value=0 (therefore rejecting $H_0$ for any signification).
And as a final example, let's imagine we are tossing a dime. We can count how many times we see the head or Franklin Roosevelt to decide if it's a fair coin and accept it's fair under a range of proportions, but if we see the image of the Queen of England or the coat of arms of Thailand on the coin, we will reject ipso facto that the coin is a dime.
