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I am trying to investigate if the proportion of successes in my population can be shown to be larger than zero. Thus, $H_0: p = 0$ and $H_A: p>0$. Since $p=0$, the prop.test in R does not work, but the bionomial test does return satisfactory results, with a p-value of $1$.

However, my main concern is the possibility of the Type II error. Given that both $p=0$ and all of my data is zero, leaving $\bar{x}=0$ and $\sigma = 0$, I am unclear of how to perform the power analysis.

I did reference this post (Power analysis for binomial data when the null hypothesis is that $p = 0$), but I don't think it has the same problem as my data presents and thus I would appreciate any direction.

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    $\begingroup$ You don't need or want any standard test for this. The observation of a single success will reject the null (with absolutely no uncertainty!) and observing all failures obviously should not reject the null. That's such a simple test that the power analysis will be straightforward and easy. $\endgroup$ – whuber Apr 17 '15 at 20:59

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