A city's Parks Department wants to estimate the percent of city residents who are satisfied with the quality of the...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
A city's Parks Department wants to estimate the percent of city residents who are satisfied with the quality of the city's parks. A random sample of \(\mathrm{340}\) city residents were surveyed. Based on the sample, it is estimated that \(\mathrm{57\%}\) of all city residents are satisfied with the parks, with an associated margin of error of \(\mathrm{4.5\%}\). Based on the estimate and associated margin of error, which of the following is the most appropriate conclusion about all city residents?
\(4.5\%\) of the residents are satisfied with the parks.
It is plausible that between \(52.5\%\) and \(61.5\%\) of the residents are satisfied with the parks.
\(57\%\) of the residents are satisfied with the parks.
It is plausible that more than \(61.5\%\) of the residents are satisfied with the parks.
1. TRANSLATE the problem information
- Given information:
- Sample of 340 residents surveyed
- Sample estimate: \(57\%\) satisfied with parks
- Margin of error: \(4.5\%\)
- What this tells us: We have a sample statistic (\(57\%\)) that estimates a population parameter, with an associated uncertainty (\(4.5\%\))
2. INFER what margin of error means
- The margin of error creates a confidence interval around our sample estimate
- This interval shows the range of plausible values for the true population percentage
- Formula: \(\mathrm{Confidence\ Interval = Sample\ Estimate ± Margin\ of\ Error}\)
3. Calculate the confidence interval
- Lower bound = \(57\% - 4.5\% = 52.5\%\)
- Upper bound = \(57\% + 4.5\% = 61.5\%\)
- Confidence interval: \([52.5\%, 61.5\%]\)
4. INFER the correct interpretation
- The interval \([52.5\%, 61.5\%]\) represents all plausible values for the true population percentage
- We cannot say the exact percentage is \(57\%\) - that's just our sample estimate
- We cannot say percentages outside this range are plausible
5. APPLY CONSTRAINTS to eliminate incorrect choices
- Choice A confuses margin of error with the actual percentage satisfied
- Choice C treats the sample estimate as the exact population value
- Choice D suggests values outside our confidence interval are plausible
- Choice B correctly identifies our calculated confidence interval
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Conceptual confusion about sample vs population: Students often don't distinguish between what the sample shows (\(57\%\) satisfied) and what we can conclude about the entire population. They think the sample percentage IS the population percentage.
This leads them to select Choice C (\(57\%\) of the residents are satisfied) because they interpret the sample result as the definitive answer about all residents.
Second Most Common Error:
Weak TRANSLATE reasoning: Students misunderstand what "margin of error" means, thinking it represents a percentage of people rather than a measure of uncertainty around the estimate.
This may lead them to select Choice A (\(4.5\%\) of the residents are satisfied) because they confuse the margin of error with the actual satisfaction percentage.
The Bottom Line:
This problem tests whether students understand that sample statistics estimate population parameters with uncertainty. The key insight is that margin of error creates a range of plausible values, not a single definitive answer.
\(4.5\%\) of the residents are satisfied with the parks.
It is plausible that between \(52.5\%\) and \(61.5\%\) of the residents are satisfied with the parks.
\(57\%\) of the residents are satisfied with the parks.
It is plausible that more than \(61.5\%\) of the residents are satisfied with the parks.