From a population of 50,000 people, 1,000 were chosen at random and surveyed about a proposed piece of legislation. Based...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
From a population of \(50,000\) people, \(1,000\) were chosen at random and surveyed about a proposed piece of legislation. Based on the survey, it is estimated that \(35\%\) of people in the population support the legislation, with an associated margin of error of \(3\%\). Based on these results, which of the following is a plausible value for the total number of people in the population who support the proposed legislation?
\(\mathrm{350}\)
\(\mathrm{650}\)
\(\mathrm{16{,}750}\)
\(\mathrm{31{,}750}\)
Step-by-Step Solution
1. TRANSLATE the survey results
- Given information:
- Total population: 50,000 people
- Sample size: 1,000 people surveyed
- Estimated support: 35%
- Margin of error: 3%
2. INFER what margin of error means
- Margin of error creates a range around the estimate
- The true percentage likely falls between:
- Lower bound: \(\mathrm{35\% - 3\% = 32\%}\)
- Upper bound: \(\mathrm{35\% + 3\% = 38\%}\)
3. SIMPLIFY to find the population range
- Apply the percentage bounds to the total population:
- Minimum supporters: \(\mathrm{50,000 \times 0.32 = 16,000}\) people
- Maximum supporters: \(\mathrm{50,000 \times 0.38 = 19,000}\) people
- Any number between 16,000 and 19,000 is plausible
4. APPLY CONSTRAINTS to select the answer
- Check which choice falls within \(\mathrm{[16,000, 19,000]}\):
- A. 350: Way below range
- B. 650: Way below range
- C. 16,750: ✓ Within range
- D. 31,750: Above range
Answer: C. 16,750
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret what the numbers represent, confusing sample size with population calculations.
They might calculate 35% of the sample (1,000) instead of the population (50,000), getting 350 supporters. This may lead them to select Choice A (350).
Second Most Common Error:
Missing conceptual knowledge about margin of error: Students don't understand that margin of error creates a range of plausible values.
They simply calculate \(\mathrm{35\% \times 50,000 = 17,500}\) and look for the closest answer, potentially selecting Choice D (31,750) if they make arithmetic errors or Choice C (16,750) by luck without proper reasoning.
The Bottom Line:
This problem tests understanding of statistical concepts (margin of error) combined with percentage calculations on population data. Success requires distinguishing between sample and population, and recognizing that survey results provide ranges, not exact values.
\(\mathrm{350}\)
\(\mathrm{650}\)
\(\mathrm{16{,}750}\)
\(\mathrm{31{,}750}\)