A park ranger asked a random sample of visitors how far they hiked during their visit. Based on the responses,...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
A park ranger asked a random sample of visitors how far they hiked during their visit. Based on the responses, the estimated mean was found to be \(\mathrm{4.5\ miles}\), with an associated margin of error of \(\mathrm{0.5\ miles}\). Which of the following is the best conclusion from these data?
It is likely that all visitors hiked between 4 and 5 miles.
It is likely that most visitors hiked exactly 4.5 miles.
It is not possible that any visitor hiked less than 3 miles.
It is plausible that the mean distance hiked for all visitors is between 4 and 5 miles.
1. TRANSLATE the problem information
- Given information:
- Random sample of park visitors asked about hiking distance
- Sample mean = \(\mathrm{4.5}\) miles
- Margin of error = \(\mathrm{0.5}\) miles
- What we're looking for: The best conclusion from this data
2. INFER what margin of error means
- Margin of error creates a confidence interval around the sample mean
- This interval estimates where the true population mean likely falls
- Confidence interval = Sample mean ± Margin of error = \(\mathrm{4.5 \pm 0.5 = [4.0, 5.0]}\) miles
3. INFER what this interval tells us
- The interval \(\mathrm{[4.0, 5.0]}\) represents plausible values for the population mean
- This is about the average for ALL visitors (population), not individual visitors
- We cannot make claims about individual hiking distances from this information
4. Evaluate each answer choice
- Choice A: Claims all individuals hiked 4-5 miles → Wrong (about individuals, not population mean)
- Choice B: Claims most hiked exactly 4.5 miles → Wrong (confuses sample mean with individual values)
- Choice C: Claims no one hiked less than 3 miles → Wrong (no information about individual minimums)
- Choice D: Claims population mean is plausibly between 4-5 miles → Correct (exactly what confidence interval tells us)
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students confuse what margin of error tells us about individuals vs. the population mean.
Many students think "\(\mathrm{4.5 \pm 0.5}\)" means that all or most individual visitors hiked between 4-5 miles. They don't understand that statistical inference is about estimating population parameters (like the mean), not about individual data points. The confidence interval tells us where the average for all visitors likely falls, not where individual visitors' distances fall.
This may lead them to select Choice A (all visitors hiked between 4 and 5 miles).
Second Most Common Error:
Conceptual confusion about sample mean: Students think the sample mean represents what most people actually did.
They reason that since the sample mean is 4.5 miles, most visitors must have hiked exactly or very close to 4.5 miles. They don't understand that a mean is calculated from varied individual values and doesn't represent what most people actually experienced.
This may lead them to select Choice B (most visitors hiked exactly 4.5 miles).
The Bottom Line:
This problem tests whether students understand the fundamental purpose of statistical inference - using sample data to make claims about population parameters, not about individuals within that population.
It is likely that all visitors hiked between 4 and 5 miles.
It is likely that most visitors hiked exactly 4.5 miles.
It is not possible that any visitor hiked less than 3 miles.
It is plausible that the mean distance hiked for all visitors is between 4 and 5 miles.