Is it possible to make probability statements about the true mean? I have seen many answers on how to interpret confidence intervals, thus I think the following reasoning is wrong. But what is the mistake?
The sample mean of 1000 samples is $\hat \mu$, the sample standard deviation is $\hat s$. Let $\mu$ be the true mean. $\phi$ is the standard normal CDF.
With the central limit theorem:
$$P\left( \mu\leq\hat \mu\right)= P\left( \frac{\mu-\hat\mu}{s}\leq 0\right) \to \phi\left( 
0 \right) = 0.5$$
Thus, given the data we have observed, the probability that $\mu$ is lesser or equal to $\hat \mu$ is approx. 0.5. (which intuitively makes a lot of sense to me)
How would I have to rephrase the result to get the interpretation right? Something like "given I repeat the experiment, the probability that $(-\infty, \hat \mu]$ covers the true mean is 0.5"?
 A: The struggle with writing $P(\mu\le\hat{\mu})$ or "the probability that $\mu$ is lesser or equal to $\hat{\mu}$" is in the wording: it appears to imply that $\mu$ is a random variable, when in fact it is a fixed value (the true mean).  The random variable is $\hat{\mu}$ (or any intervals constructed around it), so any statements about probability should make it clear enough that the probability refers to $\hat{\mu}$.  
The trickier part is once you calculate $\hat{\mu}$ from data.  This is no longer a random variable but rather a realization of a random variable, which is just another number. Suppose your sample mean is 3.4.  Then, it's inappropriate to say "there is a 50% probability that the true mean is less than or equal to 3.4" (because the true mean is not random; it either is or isn't less than 3.4), but you can say "we are 50% confident the true mean is less than or equal to 3.4", or "the sample mean is 3.4. If we were to repeat this experiment over and over again, calculating the sample mean, then the true mean would be below the sample mean 50% of the time."
