How does a confidence interval typically perform in relation to the true population mean?

A confidence interval provides a range of values that, based on the sample data, is likely to contain the true population mean. The correct understanding is that a confidence interval may or may not contain the true mean because it is based on the sample from the population and is subject to sampling variability.

When a confidence interval is calculated, it allows researchers to express the uncertainty around the estimate of the population mean. For example, in a 95% confidence interval, this indicates that if you were to take many samples and construct a confidence interval from each of them, approximately 95% of those intervals would contain the true population mean. However, for any single sample, there is still a 5% probability that the true mean is outside of the interval. This uncertainty is inherent to the nature of statistical inference.

While the confidence interval provides a probabilistic measure, it does not guarantee that the interval will successfully contain the true mean with each sample, acknowledging real-world sampling variations and limitations. This crucial point makes the second option the most accurate representation of how confidence intervals work concerning the true population mean.