In the context of constructing confidence intervals, how does decreasing the confidence level (for example, from $99\%$ to $95\%$) affect the sample size required to achieve a fixed margin of error?

• A. The required sample size remains the same.
• B. The required sample size increases.
• C. The required sample size decreases.
• D. The required sample size first increases, then decreases.

Answer: (C)

Verified step by step guidance

1. Recall the formula for the required sample size \(n\) when estimating a population mean with a fixed margin of error \(E\): \[n = \left( \frac{Z_{\alpha/2} \cdot \sigma}{E} \right)^2\] where \(Z_{\alpha/2}\) is the critical value corresponding to the confidence level, \(\sigma\) is the population standard deviation, and \(E\) is the margin of error.
2. Understand that the critical value \(Z_{\alpha/2}\) depends on the confidence level. A higher confidence level (e.g., 99%) corresponds to a larger \(Z_{\alpha/2}\), while a lower confidence level (e.g., 95%) corresponds to a smaller \(Z_{\alpha/2}\).
3. Since the margin of error \(E\) and population standard deviation \(\sigma\) are fixed, the only changing factor in the sample size formula is \(Z_{\alpha/2}\). Decreasing the confidence level reduces \(Z_{\alpha/2}\).
4. Because \(n\) is proportional to the square of \(Z_{\alpha/2}\), a smaller critical value leads to a smaller required sample size to maintain the same margin of error.
5. Therefore, decreasing the confidence level from 99% to 95% decreases the required sample size needed to achieve the fixed margin of error.