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

• A. It guarantees that the true population mean is within the interval for the current sample.
• B. It is an interval that contains $95\%$ of the sample data values.
• C. It means there is a $95\%$ probability that the true population mean is exactly at the center of the interval.
• D. It is an interval calculated from sample data that, in repeated sampling, would contain the true population mean in about $95\%$ of samples.

Answer: (D)

Verified step by step guidance

1. Understand that a 95% confidence interval (CI) 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 will contain the true population mean if we were to take many samples and build intervals each time.
3. Note that the confidence interval does NOT guarantee the true mean is within the interval for the current sample; rather, it reflects the reliability of the method over many samples.
4. Recall that the confidence interval is not about containing a certain percentage of sample data values, but about capturing the population parameter (mean) with a specified level of confidence.
5. Understand that the probability statement applies to the process of interval estimation before the sample is taken, not to the true mean being at a specific point within the interval after it is calculated.