To estimate the proportion of a population that has a certain characteristic, a random sample was selected from the population....
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
To estimate the proportion of a population that has a certain characteristic, a random sample was selected from the population. Based on the sample, it is estimated that the proportion of the population that has the characteristic is \(\mathrm{0.49}\), with an associated margin of error of \(\mathrm{0.04}\). Based on this estimate and margin of error, which of the following is the most appropriate conclusion about the proportion of the population that has the characteristic?
It is plausible that the proportion is between \(0.45\) and \(0.53\).
It is plausible that the proportion is less than \(0.45\).
The proportion is exactly \(0.49\).
It is plausible that the proportion is greater than \(0.53\).
1. TRANSLATE the problem information
- Given information:
- Population proportion estimate: 0.49
- Associated margin of error: 0.04
- What this tells us: We have a point estimate with uncertainty bounds
2. INFER the approach
- The margin of error tells us how far the true population proportion could reasonably be from our sample estimate
- We need to create an interval by going both directions from the estimate
- This gives us the range of plausible values for the true proportion
3. SIMPLIFY to find the interval bounds
- Lower bound: \(\mathrm{0.49 - 0.04 = 0.45}\)
- Upper bound: \(\mathrm{0.49 + 0.04 = 0.53}\)
- Plausible interval: \(\mathrm{[0.45, 0.53]}\)
4. APPLY CONSTRAINTS to evaluate answer choices
- A: "Between 0.45 and 0.53" ✓ (exactly matches our interval)
- B: "Less than 0.45" ✗ (outside our plausible range)
- C: "Exactly 0.49" ✗ (ignores the margin of error completely)
- D: "Greater than 0.53" ✗ (outside our plausible range)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Conceptual confusion about margin of error: Students may not understand that margin of error creates a range of plausible values. They might think the estimate itself is definitive and select Choice C (exactly 0.49), missing the fundamental concept that sampling introduces uncertainty.
Second Most Common Error:
Weak INFER skill: Students may correctly calculate the interval bounds but fail to connect this to the answer choices. They might misinterpret what "plausible" means statistically, leading to confusion between what's inside versus outside the confidence interval. This could cause them to select Choice B or D by misunderstanding which values are actually supported by the data.
The Bottom Line:
This problem tests whether students understand that statistical estimates come with uncertainty. The key insight is recognizing that margin of error defines boundaries around an estimate, and only values within those boundaries are considered plausible given the sample data.
It is plausible that the proportion is between \(0.45\) and \(0.53\).
It is plausible that the proportion is less than \(0.45\).
The proportion is exactly \(0.49\).
It is plausible that the proportion is greater than \(0.53\).