Suppose a chart shows that a $95\%$ confidence interval for the mean height of a population is $(65,70)$ inches. Which explanation is best supported by these numbers?

• A. We are $95\%$ confident that the true mean height of the population is between $65$ and $70$ inches.
• B. The true mean height is exactly $67.5$ inches.
• C. $95\%$ of the population has a height between $65$ and $70$ inches.
• D. There is a $95\%$ probability that the sample mean is between $65$ and $70$ inches.

Answer: (A)

Verified step by step guidance

1. Understand what a confidence interval represents: it is a range of values, derived from sample data, that is likely to contain the true population mean with a certain level of confidence (in this case, 95%).
2. Interpret the given interval (65, 70) inches as the range calculated from the sample data where we expect the true population mean height to lie with 95% confidence.
3. Recognize that the statement 'We are 95% confident that the true mean height of the population is between 65 and 70 inches' correctly reflects the meaning of a 95% confidence interval.
4. Note that the true mean is not necessarily exactly the midpoint (67.5 inches) of the interval; the interval is an estimate, not a precise value.
5. Understand that the confidence interval does not imply that 95% of individual heights fall within this range, nor does it mean there is a 95% probability that the sample mean lies within this interval after the data is collected.