A polling agency recently surveyed 1,000 adults who were selected at random from a large city and asked each of...

1. TRANSLATE the survey information

• Given information:
  • 1,000 adults randomly selected from a large city
  • 78% of them were satisfied with air quality
  • We need to determine which statements must be true

• What this tells us: We have sample data that we're using to evaluate claims about broader populations

2. INFER the key statistical principle

• The critical word here is 'must' - we need absolute certainty, not just probability
• Random samples give us estimates about populations, but they don't guarantee exact matches
• We need to test each statement against this principle

3. INFER the validity of each statement

Statement I Analysis:

• Claims: Exactly 78% of ALL adults in the city are satisfied
• Reality: Our sample suggests this, but samples have variability
• Conclusion: NOT necessarily true

Statement II Analysis:

• Claims: Another random sample would yield exactly 78%
• Reality: Different samples from the same population often give different results
• Conclusion: NOT necessarily true

Statement III Analysis:

• Claims: A different city would yield exactly 78%
• Reality: Different cities have different conditions and populations
• Conclusion: NOT necessarily true

4. APPLY CONSTRAINTS to select the final answer

• Since none of the statements must be true, the answer is 'None'

Answer: A. None

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning about statistical certainty: Students often confuse 'likely to be true' with 'must be true.' They see 78% as a reasonable estimate and think Statement I is obviously correct, not recognizing that samples estimate but don't perfectly represent populations.

This may lead them to select Choice C (I and II only) or Choice B (II only)

Second Most Common Error:

Conceptual confusion about sampling variability: Students don't understand that different random samples can yield different results, even from the same population. They think random sampling guarantees consistent results.

This causes them to accept Statement II as true and may lead them to select Choice B (II only)

The Bottom Line:

This problem tests understanding of a fundamental statistical concept: the difference between sample estimates and population certainties. The key insight is recognizing that 'must be true' sets a much higher bar than 'probably true' in statistical contexts.