Based on the statistical model you developed, which of the following statements about confidence intervals is true?

• A. A wider confidence interval always indicates a more precise estimate of the population parameter.
• B. The confidence level of an interval depends only on the sample mean and not on the sample size.
• C. A $95\%$ confidence interval means that if we repeated the sampling process many times, approximately $95\%$ of the calculated intervals would contain the true population parameter.
• D. A $95\%$ confidence interval means there is a $95\%$ probability that the true population parameter is within the calculated interval for this sample.

Answer: (C)

Verified step by step guidance

1. Understand the concept of a confidence interval: it is a range of values, derived from sample data, that is likely to contain the true population parameter with a certain confidence level.
2. Recall that the width of a confidence interval is influenced by the variability in the data, the sample size, and the chosen confidence level. A wider interval generally indicates less precision, not more.
3. Recognize that the confidence level (e.g., 95%) represents the long-run proportion of confidence intervals that will contain the true parameter if we repeat the sampling process many times, not a probability about a single interval.
4. Note that the confidence level does not depend solely on the sample mean; it also depends on the sample size and variability, which affect the standard error and thus the interval width.
5. Interpret the correct statement: a 95% confidence interval means that if we were to take many samples and build intervals in the same way, about 95% of those intervals would contain the true population parameter.