A company that produces socks wants to estimate the percent of the socks produced in a typical week that are...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
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 \(\mathrm{310}\) socks produced in a certain week were inspected. Based on the sample, it is estimated that \(\mathrm{12\%}\) of all socks produced by the company in this week are defective, with an associated margin of error of \(\mathrm{3.62\%}\). 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?
\(\mathrm{3.62\%}\) of the socks are defective.
It is plausible that between \(\mathrm{8.38\%}\) and \(\mathrm{15.62\%}\) of the socks are defective.
\(\mathrm{12\%}\) of the socks are defective.
It is plausible that more than \(\mathrm{15.62\%}\) of the socks are defective.
1. TRANSLATE the problem information
- Given information:
- Sample estimate: \(12\%\) of socks are defective
- Margin of error: \(3.62\%\)
- Need conclusion about all socks produced
- What this tells us: We have a sample-based estimate with uncertainty
2. INFER the approach
- The margin of error tells us the estimate has uncertainty
- We need to find the range of plausible values, not treat \(12\%\) as exact
- Confidence interval formula: estimate ± margin of error
3. SIMPLIFY the interval calculation
- Lower bound: \(12\% - 3.62\% = 8.38\%\)
- Upper bound: \(12\% + 3.62\% = 15.62\%\)
- Plausible range: between \(8.38\%\) and \(15.62\%\)
4. APPLY CONSTRAINTS to select the correct conclusion
- Choice A: Wrong - states \(3.62\%\) are defective (confuses margin of error with defect rate)
- Choice B: Correct - matches our calculated interval \([8.38\%, 15.62\%]\)
- Choice C: Wrong - treats \(12\%\) as exact (ignores uncertainty)
- Choice D: Wrong - claims values above \(15.62\%\) are plausible (exceeds our interval)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that the margin of error creates a range of uncertainty around the sample estimate.
They see "\(12\%\) defective" and think this is the definitive answer about the population, completely ignoring the margin of error. This leads them to select Choice C (\(12\%\) of the socks are defective) as if the sample perfectly represents the population with no uncertainty.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret what the margin of error represents.
They might think the margin of error itself is the defect rate, or they get confused about how to use it mathematically. This conceptual confusion about the role of margin of error leads to selecting Choice A (\(3.62\%\) of the socks are defective) or causes them to get stuck and guess randomly.
The Bottom Line:
This problem tests whether students understand that sample estimates come with uncertainty, and that margin of error quantifies the range of plausible population values. The key insight is moving from a point estimate to an interval of plausible values.
\(\mathrm{3.62\%}\) of the socks are defective.
It is plausible that between \(\mathrm{8.38\%}\) and \(\mathrm{15.62\%}\) of the socks are defective.
\(\mathrm{12\%}\) of the socks are defective.
It is plausible that more than \(\mathrm{15.62\%}\) of the socks are defective.