In the context of estimating a population parameter, how does decreasing the confidence level affect the sample size required to achieve a fixed margin of error?

• A. Decreasing the confidence level increases the required sample size.
• B. Decreasing the confidence level always results in a sample size of $1$.
• C. Decreasing the confidence level decreases the required sample size.
• D. Decreasing the confidence level does not affect the required sample size.

Answer: (C)

Verified step by step guidance

1. Recall the formula for the margin of error (ME) in estimating a population parameter, which is given by: \[ ME = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \] where \(z_{\alpha/2}\) is the critical value corresponding to the confidence level, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
2. Understand that the critical value \(z_{\alpha/2}\) depends on the confidence level: a higher confidence level corresponds to a larger \(z_{\alpha/2}\), and a lower confidence level corresponds to a smaller \(z_{\alpha/2}\).
3. Since the margin of error is fixed, rearrange the formula to solve for the sample size \(n\): \[ n = \left( \frac{z_{\alpha/2} \times \sigma}{ME} \right)^2 \]
4. Notice from the formula that the sample size \(n\) is proportional to the square of the critical value \(z_{\alpha/2}\). Therefore, if the confidence level decreases, \(z_{\alpha/2}\) decreases, which in turn decreases the required sample size \(n\) to maintain the same margin of error.
5. Conclude that decreasing the confidence level decreases the required sample size to achieve a fixed margin of error, because a lower confidence level means a smaller critical value and thus a smaller sample size is needed.