A city agency sent a survey to a random sample of 500 residents out of the city's total of 80,000...

1. TRANSLATE the problem information

• Given information:
  • Total city residents: 80,000
  • Surveys sent: 500 residents
  • Surveys completed and returned: 400 residents
  • Of those who returned surveys, 240 use public transportation ≥3 times/week

• What this tells us: We need to estimate the total number of city residents who use public transportation frequently based on survey results

2. INFER which sample size to use

• The key insight: Only use data from residents who actually responded to the survey
• We have reliable data from 400 respondents (not the 500 who were originally sent surveys)
• These 400 respondents form our sample for calculating the proportion

3. SIMPLIFY to find the proportion

• Proportion of respondents who use public transportation = \(\frac{240}{400}\)
• \(\frac{240}{400} = \frac{24}{40} = \frac{12}{20} = \frac{3}{5} = 0.6 = 60\%\)

4. INFER how to estimate the population total

• If 60% of our random sample uses public transportation frequently, we estimate that 60% of the entire city population does too
• Apply this proportion to the total population: \(0.6 \times 80,000\)

5. Calculate the final estimate

• \(0.6 \times 80,000 = 48,000\) residents

Answer: C. 48,000

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Using the wrong sample size for calculating the proportion

Students often use all 500 residents who were sent surveys rather than focusing on the 400 who actually responded. They calculate: \(\frac{240}{500} = 0.48\), then \(0.48 \times 80,000 = 38,400\).

This may lead them to select Choice B (38,400)

Second Most Common Error:

Poor TRANSLATE execution: Misunderstanding what population the estimate should represent

Some students get confused about whether they're estimating based on all residents sent surveys, all residents who responded, or some other group. This conceptual confusion can lead to using incorrect numbers in their calculations.

This leads to confusion and guessing among the remaining answer choices.

The Bottom Line:

The critical insight is recognizing that statistical estimates should be based only on people who actually provided data (the 400 respondents), not on the larger group that was contacted but didn't respond (the original 500).