How does increasing the sample size $n$ affect the width of the confidence interval for the population mean $\mu$?

• A. The width of the confidence interval first increases and then decreases as sample size $n$ increases.
• B. The width of the confidence interval decreases as the sample size $n$ increases.
• C. The width of the confidence interval increases as the sample size $n$ increases.
• D. The width of the confidence interval remains the same regardless of sample size $n$.

Answer: (B)

Verified step by step guidance

1. Recall the formula for the confidence interval (CI) for a population mean when the population standard deviation is known or the sample size is large: \[ CI = \bar{x} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \] where \(\bar{x}\) is the sample mean, \(z_{\alpha/2}\) is the critical z-value for the desired confidence level, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
2. Focus on the margin of error (the part that determines the width of the confidence interval): \[ \text{Margin of Error} = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \]
3. Notice that the margin of error depends inversely on the square root of the sample size \(n\). This means as \(n\) increases, the denominator \(\sqrt{n}\) increases, making the margin of error smaller.
4. Since the width of the confidence interval is twice the margin of error (because it extends equally on both sides of the sample mean), increasing the sample size \(n\) results in a narrower confidence interval.
5. Therefore, increasing the sample size decreases the width of the confidence interval, making our estimate of the population mean more precise.