Which of the following best describes a $95\%$ confidence interval for a population mean?

• A. $95\%$ of the data values in the sample fall within the confidence interval.
• B. The true population mean will always fall within the calculated interval.
• C. If we were to take many random samples and construct a confidence interval from each sample, about $95\%$ of those intervals would contain the true population mean.
• D. There is a $95\%$ probability that the true population mean is equal to the sample mean.

Answer: (C)

Verified step by step guidance

1. Understand that a 95% confidence interval for a population mean is constructed from sample data to estimate the range in which the true population mean likely falls.
2. Recognize that the confidence level (95%) refers to the long-run proportion of such intervals that would contain the true population mean if we repeated the sampling process many times.
3. Note that the confidence interval does not guarantee that the true mean is within any single calculated interval, nor does it describe the probability of the mean being equal to the sample mean.
4. Also, understand that the confidence interval is about the parameter (population mean), not about the distribution of individual data points in the sample.
5. Therefore, the best interpretation is: If we were to take many random samples and construct a confidence interval from each sample, about 95% of those intervals would contain the true population mean.