Residents of a town were surveyed to determine whether they are satisfied with the concession stand at the local park....

1. TRANSLATE the sampling scenario

• Given information:
  • Random sample of 200 residents surveyed
  • \(\mathrm{87\%}\) of these 200 residents said they are satisfied
  • Need to determine what MUST be true
• What this tells us: We have sample data, not complete population data

2. INFER what "must be true" means in statistics

• "Must be true" means absolutely certain based on the given information
• Just because something could happen or seems likely doesn't mean it must happen
• We need to distinguish between sample results and population truths

3. INFER the validity of Statement I

• Statement I claims: "Of all town residents, \(\mathrm{87\%}\) would say they are satisfied"
• This assumes sample percentage = population percentage
• Key insight: Samples estimate populations but don't necessarily match exactly
• Conclusion: Statement I is NOT necessarily true

4. INFER the validity of Statement II

• Statement II claims: "Another sample of 200 would give \(\mathrm{87\%}\) satisfied"
• This assumes all samples yield identical results
• Key insight: Sampling variability means different samples can give different results
• Conclusion: Statement II is NOT necessarily true

5. Select the answer

• Neither statement must be true
• Answer: A. Neither

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students confuse "reasonable" with "must be true"

They think: "\(\mathrm{87\%}\) seems like a good estimate, so probably close to \(\mathrm{87\%}\) of all residents are satisfied" and select Choice B. Or they think "another similar sample would probably give similar results" and get confused between choices.

The key misconception is not understanding that statistics problems often ask what MUST be true (certainty) rather than what's likely to be true (probability). Sample results, no matter how good, don't guarantee identical population parameters or identical future samples.

This leads them to select Choice B (I only) or Choice D (I and II).

The Bottom Line:

This problem tests understanding of sampling limitations - samples provide estimates, not guarantees. The critical skill is recognizing the difference between statistical likelihood and logical necessity.