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The problem describes a company inspecting socks for defects. The company sampled 310 socks and found that 12% of them were defective, with a margin of error of 3.62%. The goal is to determine the most appropriate conclusion based on this data.

### Key information:
- **Estimate of defective socks**: 12%
- **Margin of error**: 3.62%

The margin of error tells us the range of plausible values for the true percentage of defective socks. To calculate this range:
- Lower bound = \( 12\% - 3.62\% = 8.38\% \)
- Upper bound = \( 12\% + 3.62\% = 15.62\% \)

This means that based on the sample, the true percentage of defective socks could plausibly be between **8.38% and 15.62%**.

### Let's analyze the answer choices:
- **A. 3.62% of the socks are defective**: This is incorrect because 3.62% is the margin of error, not the percentage of defective socks.
- **B. It is plausible that between 8.38% and 15.62% of the socks are defective**: This is correct because the true percentage could fall within this range.
- **C. 12% of the socks are defective**: This is incorrect because 12% is just the point estimate, not necessarily the true percentage.
- **D. It is plausible that more than 15.62% of the socks are defective**: This is incorrect because 15.62% is the upper bound of the margin of error, so it's unlikely the percentage is higher than this.

### Correct answer: **B**

Do you have any further questions or need details?

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Here are 5 follow-up questions to deepen your understanding:
1. How is the margin of error calculated in statistical sampling?
2. What factors can influence the size of the margin of error in a study?
3. Why is it important to interpret confidence intervals, not just point estimates, in statistics?
4. How does increasing the sample size affect the margin of error?
5. What would happen to the range of plausible values if the margin of error were smaller?

**Tip**: Confidence intervals provide a range of values that likely contain the true parameter. Always consider both the estimate and the margin of error for better decision-making.

A company that produces socks wants to estimate the percent of the socks produced in a typical week that are defective. A random sample of 310 socks produced in a certain week were inspected. Based on the sample, it is estimated that 12% of all socks produced by the company in this week are defective, with an associated margin of error of 3.62%. Based on the estimate and associated margin of error, which of the following is the most appropriate conclusion about all socks produced by the company during this week?

A) 3.62% of the socks are defective.
B) It is plausible that between 8.38% and 15.62% of the socks are defective.
C) 12% of the socks are defective.
D) It is plausible that more than 15.62% of the socks are defective.

Learn how to estimate the percentage of defective socks produced by a company using a sample. This problem involves calculating the margin of error and interpreting confidence intervals. Suitable for students in Grades 9-12.

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