Working on Confidence Interval and its meaning [duplicate]

Supposed in a population, I take a sample with a size of 1500 to build a 96% confidence interval for the true mean $\mu$. Then such an interval is assumed to range from 6.8 to 8.0. Assume the sample mean is $\overline{x}$. Then I conclude that:

if a sample with a size of 1500 were repeatedly constructed, then over a long run 96% of the confidence interval formed would contain the true mean $\mu$. (Another trivial interpretation is that I am 96% confident that the true mean will lie between 6.8 and 8.0 so I don't want to bother to mention this interpretation).

Am I correct with my above conclusion? Thank you!

marked as duplicate by Stephan Kolassa, mdewey, kjetil b halvorsen, Peter Flom♦Nov 22 '17 at 13:18

In a frequentist paradigm we treat the true mean $\mu$ as fixed. Since we are taking a sample from the generating distribution the randomness comes from this sample. In this way we construct a confidence interval. In this way, your first interpretation is correct: if we were to sample 1500 observations from the same distribution repeatedly then a confidence level of $\alpha$ implies that the true parameter $\mu$ would fall within the confidence interval $\alpha$% of the time.
The second, seemingly more natural interpretation is incorrect from a frequentist perspective. It is more akin to the credible interval of the Bayesian paradigm. Recall that a Bayesian treats the parameter $\mu$ as random. An $\alpha$% credible interval of [a, b] suggests that the subjective (due to the observed data) probability that $\mu \in [a, b]$ is $\alpha$%.