Which of the following is a primary benefit of having a large sample size when constructing confidence intervals?

• A. A large sample size results in a narrower confidence interval, providing a more precise estimate of the population parameter.
• B. A large sample size guarantees that the sample mean $ṅ$ is exactly equal to the population mean $\mu$.
• C. A large sample size increases the confidence level without changing the interval width.
• D. A large sample size eliminates all sampling error.

Answer: (A)

Verified step by step guidance

1. Understand that a confidence interval estimates a population parameter (like a mean) by providing a range of plausible values based on sample data.
2. Recall that the width of a confidence interval depends on the standard error, which is calculated as \(\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.
3. Recognize that as the sample size \(n\) increases, the denominator \(\sqrt{n}\) increases, causing the standard error to decrease.
4. Since the confidence interval width is proportional to the standard error, a smaller standard error results in a narrower confidence interval.
5. Conclude that a narrower confidence interval means a more precise estimate of the population parameter, which is the primary benefit of having a large sample size.