Which of the following best describes the meaning of a $99\%$ confidence interval for a population mean?

• A. If we were to take many random samples and construct a confidence interval from each, about $99\%$ of those intervals would contain the true population mean.
• B. There is a $99\%$ probability that the true population mean is equal to the sample mean.
• C. $99\%$ of the sample data values fall within the confidence interval.
• D. The true population mean will always fall within the calculated interval.

Answer: (A)

Verified step by step guidance

1. Understand that a confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter (in this case, the population mean).
2. Recognize that a 99% confidence interval means that if we were to take many random samples from the population and compute a confidence interval from each sample, approximately 99% of those intervals would contain the true population mean.
3. Note that the confidence level (99%) does not imply the probability that the true mean lies within any one specific interval calculated from a single sample; instead, it refers to the long-run proportion of such intervals that will capture the true mean.
4. Understand that the statement 'There is a 99% probability that the true population mean is equal to the sample mean' is incorrect because the population mean is a fixed value, not a random variable, and the sample mean is an estimate that varies from sample to sample.
5. Also, recognize that the other options are incorrect because 99% of the sample data values do not necessarily fall within the confidence interval, and the true population mean does not always fall within any single calculated interval, but rather within the intervals constructed from repeated sampling about 99% of the time.