If a 95% confidence interval for a mean is 10 to 20, what can be inferred?

Here's the solution for question 64:

64. If a 95% confidence interval for a mean is 10 to 20, what can be inferred?

• A. The population mean will always lie within the interval. This is incorrect. A 95% confidence interval means that if we were to repeat the sampling process many times, 95% of the intervals constructed would contain the true population mean. This implies there's a 5% chance that any given interval does not contain the true mean.
• B. The sample data is normally distributed. This is not necessarily true. While normality of the sample data or the sampling distribution of the mean is often an assumption, especially for small sample sizes, the Central Limit Theorem allows us to construct valid confidence intervals for large sample sizes even if the original data is not normally distributed. The confidence interval itself does not directly infer the distribution of the sample data.
• C. The population mean is normally distributed. This is incorrect. The population mean is a fixed parameter, not a random variable, and therefore does not have a distribution.
• D. There is a 95% probability that the population mean lies within the interval. This is the most commonly accepted interpretation in many educational contexts. It reflects the confidence level associated with the method used to construct the interval. While a strict frequentist interpretation would state "we are 95% confident that the interval contains the true population mean" (referring to the reliability of the method), this option is the closest and most practical interpretation among the choices provided. It conveys that the process of creating the interval has a 95% chance of capturing the true mean.

The final answer is $\boxed{\text{D}}$