Computing type II error $\beta$ In hypothesis testing for the value of the population mean, when computing type II error $\beta$, if the alternative hypothesis includes many possibilities, is it always the case that $\beta$ is approximated? In other words, when we compute a numerical value for $\beta$, based on the mean from the sample $\bar{X}$, shouldn't we write $\beta \approx P[\text{do not reject } H_0 \mid \mu = \bar{X} ]$? 
If there is no other information available, of course, it's the only way to compute $\beta$. It's the best we can do. But since $\bar{X}$ may very well not be $\mu$, shouldn't it be stressed that it's $\approx$ and not $=$?
Edit: the original non-answered question was here... https://math.stackexchange.com/questions/298922/computing-type-ii-error-beta
 A: Well, given that $\beta$ uses the greek symbol, as a matter of nomenclature we'd expect it to be a population value.  As a population value the number is considered to be certain (=), not an estimate ($\approx$).  Moreover, as alluded to by @StasK, the hypothesis are about the population parameters, not the sample statistics.
One problem here is with notation.  $\beta$ and $1-\beta$ (power) are relevant only in worlds where $\mu_{control} \neq \mu_{treatment}$.  Therefore, your question should be "shouldn't we write $\beta \approx P[\text{do not reject }H_0 | \mu_{control} \neq \mu_{treatment}]$".
I think some of the confusion here comes from a priori vs posthoc power analysis.  In the case of a priori analyses we make assumptions about the population parameters and thereby yield the associated $\beta$ given those assumptions.  
Although it is true that in the case of posthoc analyses we are using sample statistics to calculate $\beta$ we are implicitly considering them to be population parameters.  Therefore, although it is true that $\beta \approx P[\text{retain }H_0 | \mu = \bar{X}]$ it assumed that what we really are saying is that $\beta = P[\text{do not reject }H_0 | \mu_{control} = \mu_{treatment}]$.  Therefore, in the formulation including $\bar{X}$ the quantity being estimated should not have the $\beta$ notation, but the sample equivelant.  If you are willing to make the (common) assumption (for this type of analysis) that $\bar{X} = \mu_{treatment}$ then you are good to use $=$ rather than $\neq$.
