A researcher surveyed 30 students at random from a school of 600 students. She found that 18 of the students...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
A researcher surveyed \(\mathrm{30}\) students at random from a school of \(\mathrm{600}\) students. She found that \(\mathrm{18}\) of the students in the sample participate in exactly two extracurricular activities. Based on the researcher's findings, which of the following is the best estimate of the number of students at the school who participate in exactly two extracurricular activities?
- \(\mathrm{12}\)
- \(\mathrm{360}\)
- \(\mathrm{570}\)
- \(\mathrm{582}\)
12
360
570
582
1. TRANSLATE the problem information
- Given information:
- Total school population: 600 students
- Random sample: 30 students surveyed
- Sample result: 18 students participate in exactly two extracurricular activities
- What we need: Estimate how many students in the entire school participate in exactly two extracurricular activities
2. INFER the statistical approach
- Since this is a random sample, we can use the sample proportion to estimate the population proportion
- Key insight: What's true for the sample should approximately hold for the entire population
3. SIMPLIFY to find the sample proportion
- Sample proportion = 18 students ÷ 30 students = \(\frac{18}{30}\) = \(\frac{3}{5}\) = \(0.6\)
- This means \(60\%\) of the sample participates in exactly two activities
4. INFER and apply the proportion to the total population
- If \(60\%\) of the sample has this characteristic, we estimate \(60\%\) of the total population has it too
- Calculation: \(0.6 \times 600 = 360\) students
Answer: B (360)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't understand how sample data connects to population estimates. They might think the problem is asking them to subtract the sample from the total population, leading them to calculate \(600 - 30 = 570\), and then get confused about what to do with the 18. This may lead them to select Choice C (570) or abandon systematic solution and guess.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret what the 18 represents or how to use it. They might think 18 is the total answer, or try to add it directly to some calculation involving 600. This may lead them to select Choice A (12) if they calculate \(30 - 18 = 12\), or Choice D (582) if they try \(600 - 18 = 582\).
The Bottom Line:
This problem tests whether students understand that random samples can be used to make inferences about larger populations - a fundamental concept in statistics that requires connecting sample data to population estimates through proportional reasoning.
12
360
570
582