A polling company wants to estimate the percent of registered voters in a city who support a proposed transportation initiative....

1. TRANSLATE the problem information

• Given information:
  • Sample estimate: 37% of voters support the initiative
  • Margin of error: 5.1%
  • Need to determine what this tells us about all registered voters

2. INFER what margin of error means

• Margin of error creates a confidence interval around our sample estimate
• This gives us a range of plausible values for the true population parameter
• We calculate: sample estimate ± margin of error

3. SIMPLIFY to find the confidence interval bounds

• Lower bound: \(37\% - 5.1\% = 31.9\%\)
• Upper bound: \(37\% + 5.1\% = 42.1\%\)
• Confidence interval: \((31.9\%, 42.1\%)\)

4. INFER the correct interpretation

• The confidence interval means it's plausible that the true percentage of all registered voters who support the initiative falls anywhere between 31.9% and 42.1%
• We cannot say exactly 37% support it - that's just our sample estimate

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about sample estimates vs. population parameters: Students treat the 37% sample estimate as if it's the exact percentage of all voters who support the initiative.

They think: "The survey found 37% support, so 37% of all voters support it." This leads them to select Choice C (37% of the voters support the initiative) because they don't understand that sample estimates have uncertainty.

Second Most Common Error:

Weak TRANSLATE reasoning: Students misinterpret what the 5.1% margin of error represents, thinking it's the actual percentage of voter support rather than the uncertainty measure.

This confusion about basic statistical terminology may lead them to select Choice A (5.1% of the voters support the initiative) or causes them to get stuck and guess randomly.

The Bottom Line:

This problem tests whether students understand the fundamental concept that sample statistics are estimates with uncertainty, not exact population values. The key insight is recognizing that margin of error creates a range of plausible values, not a single definitive answer.