A city's transportation department conducted a study to estimate the mean commute time for its residents to help guide infrastructure...

1. TRANSLATE the problem information

• Given information:
  • Sample mean = 28.5 minutes
  • Plausible range (confidence interval) = 26.1 to 30.9 minutes
  • Need to find: margin of error

• What this tells us: We have a confidence interval with known bounds and center point.

2. INFER the relationship

• Key insight: A confidence interval has the structure \(\mathrm{Sample\ Mean \pm Margin\ of\ Error}\)
• This means the margin of error is the distance from the sample mean to either endpoint
• We can use either the upper bound or lower bound to find this distance

3. SIMPLIFY using the lower bound

• Formula: \(\mathrm{Sample\ Mean - Margin\ of\ Error = Lower\ Bound}\)
• Substituting: \(\mathrm{28.5 - Margin\ of\ Error = 26.1}\)
• Solving: \(\mathrm{Margin\ of\ Error = 28.5 - 26.1 = 2.4}\) minutes

4. Verify using the upper bound

• Formula: \(\mathrm{Sample\ Mean + Margin\ of\ Error = Upper\ Bound}\)
• Substituting: \(\mathrm{28.5 + Margin\ of\ Error = 30.9}\)
• Solving: \(\mathrm{Margin\ of\ Error = 30.9 - 28.5 = 2.4}\) minutes

Answer: B (2.4 minutes)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about margin of error: Students confuse the total width of the confidence interval with the margin of error itself.

They calculate: \(\mathrm{30.9 - 26.1 = 4.8}\) minutes and think this IS the margin of error, not realizing that margin of error is half the total width (the distance from center to either end, not end to end).

This leads them to select Choice C (4.8 minutes).

Second Most Common Error:

Weak TRANSLATE reasoning: Students misidentify what the margin of error represents, thinking it's the same as the sample mean.

They see 28.5 minutes as the central value and incorrectly assume this must be the margin of error, not understanding that the sample mean is the center point around which the margin of error extends.

This may lead them to select Choice D (28.5 minutes).

The Bottom Line:

Success on this problem requires clearly understanding that margin of error measures the distance from the center of a confidence interval to its edges, not the total width or the center value itself.