True or false? A larger sample size produces a longer confidence interval for $\mu$.

• A. False
• B. True

Answer: (A)

Verified step by step guidance

1. Understand what a confidence interval represents: it is a range of values, derived from the sample data, that is likely to contain the true population mean \( \mu \) with a certain level of confidence (e.g., 95%).
2. Recall the formula for a confidence interval for the population mean when the population standard deviation is known: \[ CI = \bar{x} \pm z^* \times \frac{\sigma}{\sqrt{n}} \] where \( \bar{x} \) is the sample mean, \( z^* \) is the critical value from the standard normal distribution, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
3. Focus on the margin of error part of the confidence interval: \[ ME = z^* \times \frac{\sigma}{\sqrt{n}} \] Notice that the margin of error depends inversely on the square root of the sample size \( n \).
4. Analyze how the sample size \( n \) affects the length of the confidence interval: as \( n \) increases, \( \frac{1}{\sqrt{n}} \) decreases, which makes the margin of error smaller, thus the confidence interval becomes narrower (shorter).
5. Conclude that a larger sample size produces a shorter confidence interval for \( \mu \), so the statement 'A larger sample size produces a longer confidence interval' is false.