# how do I interpret the following hypothesis test?

Let's say I have two hypotheses for a coin with probability $p$ for heads:

$H_0$ - the null hypothesis - the coin is fair $p = 0.5$.

$H_1$ - coin is unfair $p \neq 0.5$.

Say the test is $|X-n/2| > r$ for some $r$ where $n$ is the number of tosses.

It is possible to choose $r$ for a desired significance level quite easily (theoretically, all the details are there for computation).

But what if I want to design a test with a certain type II error? It seems odd to me that one cannot compute $p(\textit{accept } H_0 | H_1)$ because $H_1$ doesn't specify what is the value of $p$.

When I looked around, I noticed that for this simple example, it is requested to calculate type II error for a certain value of $p \neq 0.5$ (such as $p = 0.7$).

Is there a meaning to type II error when $p$ is unknown?

Also, how is that related to the idea of "frequentist repeated experiments?"

Thanks.

• Is your $H_1$ that p>0.5 or that it is not equal to 0.5 (as implied by your test statistic)? Commented Jan 3, 2013 at 22:49
• I guess I misunderstand something - aren't the test statistic and the hypothesis decoupled from each other? I can set $H_1$ to whatever I want, $H_0$ to whatever I want, and the test statistics to whatever I want, and calculate the errors of both kinds. Or maybe you are saying out of common sense, I should change $H_1$. Commented Jan 3, 2013 at 22:54
• I changed it to be $\neq 0.5$. Commented Jan 3, 2013 at 22:54
• You don't need $H_1$ at all in some approaches (where $H_1$ is just NOT $H_0$), but if you have one it needs to be consistent with your statistic you are using to choose between the two. The question is clearer now. Commented Jan 3, 2013 at 23:09
• Re: "Is there a meaning to type II error when p is unknown?" - well, sure, it has a meaning, but that it carries meaning doesn't imply you can compute its value. Commented Jan 3, 2013 at 23:38