I understand what confidence intervals are and how we interpret them, but I guess one thing that I never realized in understanding this concept was what exactly the single confidence interval estimate we provide actually means. So as an example, say we find a 95% confidence interval for a parameter estimate $\beta$. Say we got a point estimate $\hat{\beta}$ as well as the necessary standard error, $\text{s}(\hat{\beta})$ and critical value (we'll use a t-statistic here), $t(1-\alpha; n-p)$. So we go on to construct a confidence interval:
$$ \hat{\beta} \pm t(1-\frac{\alpha}{2}; n-p)\cdot\text{s}(\hat{\beta})$$
So the interpretation of this is that over repeated sampling, 95% of the constructed interval estimates will contain the true value of the parameter. But as you see above in practice I calculated a confidence interval explicitly. So how do we frame this with regards to the whole notion of confidence intervals? Would I say this one explicitly calculated confidence interval is one of the possible intervals that would arise 95% of the time? What is it I'm missing in the understanding of this?
I'm aware of the popular post: What, precisely, is a confidence interval? and actually I have it saved as a bookmark. The overall idea of the CI is not what I'm stuck on it is these sort of individual outputs.
