SamplePercent in favorMargin of errorA52%4.2%B48%1.6%The results of two random samples of votes for a proposition are shown above. The samples...

1. INFER what factors affect margin of error

• Given information:
  • Sample A: 52% favorable, 4.2% margin of error
  • Sample B: 48% favorable, 1.6% margin of error
  • Both samples from same population, same calculation method

• This tells us the confidence level and methodology are identical

2. INFER which factors could cause the difference

• Since margin of error formula is \(\mathrm{ME ≈ z × \sqrt{p(1-p)/n}}\), we need to check:
  • z-score (confidence level): Same for both - controlled
  • p(1-p) term: Sample A = \(\mathrm{0.52(0.48) = 0.2496}\), Sample B = \(\mathrm{0.48(0.52) = 0.2496}\) - Identical!
  • Sample size (n): This must be the differentiating factor

3. INFER the relationship direction

• Margin of error has inverse relationship with sample size
• Larger margin of error → Smaller sample size
• Since Sample A has larger margin of error (4.2% vs 1.6%), Sample A must have smaller sample size

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students focus on the percentage difference (52% vs 48%) without recognizing that both are equidistant from 50%, making their contribution to margin of error identical.

They think "higher percentage must mean higher margin of error" and select Choice B (Sample A had a higher percent of favorable responses).

Second Most Common Error:

Conceptual confusion about sample size relationship: Students incorrectly believe that larger samples create larger margins of error, thinking "more data means more uncertainty."

This leads them to select Choice C (Sample A had a larger sample size) - the exact opposite of the correct relationship.

The Bottom Line:

This problem tests understanding that margin of error primarily depends on sample size when other factors are controlled, and that the relationship is inverse - bigger samples give more precise estimates with smaller margins of error.