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 $=$?
