Which statement about a 95% confidence interval for a mean is correct?

• A. There is a 95% probability that μ lies in the interval.
• B. We are 95% confident that μ lies in the interval.
• C. The true μ is definitely inside the interval.
• D. The data are 95% likely to be normally distributed.

Answer: (B)

The main idea is what a 95% confidence interval says about the method, not about a single fixed value. If you could repeat the whole sampling process many times and compute a new confidence interval each time, about 95% of those intervals would contain the true mean. This reflects the long-run performance of the interval-estimation procedure under the assumed model.

For your current data, the interval is a random result of sampling. The true mean is a fixed quantity, so saying there is a 95% probability that the mean lies in this particular interval isn’t the right interpretation. It either does or doesn’t contain the true mean, but we don’t know which in this single study.

It also isn’t asserting that the true mean is definitely inside. There’s always some chance it isn’t, given sampling variability. And the statement isn’t about the data distribution being 95% normal either; the confidence level refers to the coverage of the interval over repeated samples, assuming the method and model are correct.

So the correct interpretation is that we are 95% confident that the interval contains the true mean, meaning the procedure used to construct the interval has 95% long-run coverage under the stated assumptions.