A quality control engineer collected a simple random sample of 10 bolts produced in a batch to assess their lengths (in millimeters). The sample has the following statistics:


Sample mean $\left(\bar{x}\right)=24.07\text{ mm}$
Margin of error (ME) for a $90\%$ confidence interval $=0.11\text{ mm}$


Construct the $90\%$ confidence interval for the true mean length of bolts produced in the first batch using the given statistics. Another confidence interval for a sample of bolts from a second batch is reported as $(23.95\text{ mm},24.25\text{ mm})$. How does the result compare to the confidence interval of the second batch? Explain possible reasons for differences in their widths.

• A. The interval is $\left(24.00\text{ mm, }24.14\text{ mm}\right)$; both intervals are equal in width due to equal sample sizes.
• B. The interval is $\left(23.50\text{ mm, }24.67\text{ mm}\right)$; the second interval is wider due to a lower confidence level.
• C. The interval is $\left(24.12\text{ mm, }24.3\text{ mm}\right)$; the second interval is narrower due to lower variability.
• D. The interval is $\left(23.96\text{ mm, }24.18\text{ mm}\right)$; the second interval is wider, likely due to higher variability or smaller sample size.

Answer: (D)