I'm learning confidence interval/level, and I was taught that confidence interval can measure how representative our sample is. But I don't know why.
For example, assuming that we are given a sample of size N from population, and after calculating, we get the sample mean $\overline x$ and the standard deviation of the mean $\sigma_\overline x$. Given the confidence level 95%, we get the confidence interval.
$$\overline x \pm1.96\sigma_\overline x$$
It means that, were this procedure to be repeated on multiple samples, the calculated confidence interval ($\overline x \pm1.96\sigma_\overline x$) would encompass the true population parameter 95% of the time.
What confuses me is that $\overline x$ is different for each sample. So the confidence interval is different for each sample. If we draw a sample, and get its parameters from calculation
$$\overline x = 5\\ \sigma_\overline x = 1$$
Can we say were this procedure to be repeated on multiple samples, the calculated confidence interval ($[5-1.96, 5+1.96]$) would encompass the true population parameter 95% of the time? I think the answer is negative, because for each sample, $\overline x$ is a different value, so the confidence interval for each sample is different. If each sample's confidence interval is different, why should we use it to denote how representative our sample is?
And if we have a confidence interval $[5-1.96, 5+1.96]$, how should I interpret the interval without using $\overline x$(because $\overline x$ varies from sample to sample)?