A researcher uses a random sample to estimate the average one-way commute time for workers in a city. The estimate...

1. TRANSLATE the problem information

• Given information:
  • Sample estimate of average commute time: 37 minutes
  • Margin of error: 4 minutes
• We need to determine what this tells us about the true population average

2. INFER the meaning of margin of error

• Margin of error creates a plausible range around our sample estimate
• The true population average is likely to fall within: \(\mathrm{estimate \pm margin\,of\,error}\)
• This gives us the interval: \([37 - 4, 37 + 4] = [33, 41]\) minutes

3. APPLY CONSTRAINTS to evaluate answer choices

• (A) Between 33 and 41 minutes → Within our interval ✓
• (B) Less than 33 minutes → Outside our interval ✗
• (C) Exactly 37 minutes → Sample estimate ≠ true mean ✗
• (D) Greater than 41 minutes → Outside our interval ✗

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about margin of error: Students may think the margin of error means "the estimate could be wrong by 4 minutes in either direction, so any value is possible" rather than understanding it defines a specific plausible interval.

This leads them to think that values like 30 minutes or 45 minutes are equally plausible, causing them to select Choice B or Choice D or to become confused and guess.

Second Most Common Error:

Weak INFER skill: Students may correctly calculate the interval \([33, 41]\) but then misunderstand what "plausible" means, thinking the estimate must be exactly correct.

This may lead them to select Choice C (exactly 37 minutes) because they focus on the point estimate rather than the interval.

The Bottom Line:

The key insight is that margin of error defines boundaries - it tells us where the true value plausibly lies, not that any value is equally likely. Values outside the margin-of-error interval are not considered plausible.