If the confidence level is increased from $90\%$ to $99\%$, what happens to the width of the confidence interval for a population mean, assuming all other factors remain constant?

• A. The width of the confidence interval becomes zero.
• B. The width of the confidence interval decreases.
• C. The width of the confidence interval stays the same.
• D. The width of the confidence interval increases.

Answer: (D)

Verified step by step guidance

1. Recall that the confidence interval for a population mean is generally given by the formula: \[ \text{CI} = \bar{x} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \] where \(\bar{x}\) is the sample mean, \(z_{\alpha/2}\) is the critical z-value corresponding to the confidence level, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
2. Understand that the width of the confidence interval is determined by the margin of error, which is: \[ \text{Margin of Error} = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \]
3. Recognize that increasing the confidence level from 90% to 99% means that \(z_{\alpha/2}\) increases because a higher confidence level requires capturing more of the distribution's area, leading to a larger critical value.
4. Since \(\sigma\) and \(n\) remain constant, the only changing factor affecting the width is \(z_{\alpha/2}\). An increase in \(z_{\alpha/2}\) results in a larger margin of error, thus increasing the width of the confidence interval.
5. Therefore, when the confidence level increases from 90% to 99%, the width of the confidence interval increases to reflect the greater certainty required.