The concept of confidence intervals

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.

The interval covers the true parameter with a probability $1-\alpha$ or the true parameter lies in the interval with a probability $1-\alpha$? The answer is the latter.

Confidence intervals consist of a range of values (interval) that act as good estimates of the unknown population parameter. However, in infrequent cases, none of these values may cover the value of the parameter. The level of confidence of the confidence interval would indicate the probability that the confidence range captures this true population parameter given a distribution of samples. It does not describe any single sample.

Function Rosling.bubbles() in the animation package shows the concept of the confidence interval which depends on the observations: if the samples change, the interval changes too. At last we can see that the coverage rate will be approximate to the confidence level.

library(animation)
ani.options(nmax = 100, interval = 0.15)
par(mar = c(3, 3, 1, 0.5), mgp = c(1.5, 0.5, 0), tcl = -0.3)
conf.int()