A company designs and makes handbags. To estimate the mean weight of the handbags made by the company on a...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
A company designs and makes handbags. To estimate the mean weight of the handbags made by the company on a particular day, a sample of the handbags made by the company on that day was selected at random. Based on the sample, it is estimated that the mean weight of all handbags made by the company on that day is \(27.8\) ounces (oz), with an associated margin of error of \(0.02\) oz. Based on this estimate and associated margin of error, which of the following is the most plausible conclusion?
The mean weight of all handbags made by the company on that day is between \(27.78\text{ oz}\) and \(27.82\text{ oz}\).
The actual weights of all handbags made by the company on that day are between \(27.78\text{ oz}\) and \(27.82\text{ oz}\).
The actual weights of all handbags from the sample are between \(27.78\text{ oz}\) and \(27.82\text{ oz}\).
The mean weight of all handbags made by the company on that day is \(27.8\text{ oz}\).
1. TRANSLATE the problem information
- Given information:
- Estimated mean weight = 27.8 oz
- Margin of error = 0.02 oz
- We need to determine what this tells us about the true mean weight
2. INFER what margin of error means
- Margin of error creates a range of plausible values around our estimate
- The true population mean is likely somewhere within this range
- This applies specifically to the MEAN weight, not individual handbag weights
3. SIMPLIFY to find the range
- Lower bound: \(\mathrm{27.8 - 0.02 = 27.78}\) oz
- Upper bound: \(\mathrm{27.8 + 0.02 = 27.82}\) oz
- So the mean weight is plausibly between 27.78 oz and 27.82 oz
4. INFER which choice matches our interpretation
- Choice A: Mean weight between 27.78 oz and 27.82 oz ✓ (This matches!)
- Choice B & C: Individual weights in this range (Wrong - margin of error doesn't apply to individual values)
- Choice D: Mean is exactly 27.8 oz (Wrong - ignores the uncertainty)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Conceptual confusion about margin of error: Students don't understand that margin of error applies to the estimated mean, not to individual data values.
They might think: "If the mean is \(\mathrm{27.8 \pm 0.02}\), then all handbags must weigh between 27.78 and 27.82 oz." This fundamental misunderstanding of what margin of error represents leads them to select Choice B (individual weights between 27.78 oz and 27.82 oz) or Choice C (sample weights between 27.78 oz and 27.82 oz).
Second Most Common Error:
Weak INFER skill about statistical uncertainty: Students treat the estimate as an exact value rather than recognizing the inherent uncertainty.
They might think: "The estimate is 27.8 oz, so that must be the exact mean weight." This ignores the purpose of reporting a margin of error and may lead them to select Choice D (mean weight is 27.8 oz).
The Bottom Line:
This problem tests whether students understand that margin of error quantifies uncertainty about a population parameter (the mean), not about individual observations. Success requires distinguishing between "what's the range for the true mean?" versus "what's the range for individual values?"
The mean weight of all handbags made by the company on that day is between \(27.78\text{ oz}\) and \(27.82\text{ oz}\).
The actual weights of all handbags made by the company on that day are between \(27.78\text{ oz}\) and \(27.82\text{ oz}\).
The actual weights of all handbags from the sample are between \(27.78\text{ oz}\) and \(27.82\text{ oz}\).
The mean weight of all handbags made by the company on that day is \(27.8\text{ oz}\).