At a large high school, 300 students were selected at random and were asked in a survey about a menu...

1. TRANSLATE the survey information

• Given information:
  • 300 students randomly surveyed
  • \(\mathrm{38\%}\) support menu change
  • Margin of error: \(\mathrm{5.5\%}\)
• What this tells us: We have a sample estimate with uncertainty bounds

2. INFER what margin of error means

• Margin of error tells us how much the true population percentage could differ from our sample estimate
• The population percentage could be \(\mathrm{5.5\%}\) higher OR \(\mathrm{5.5\%}\) lower than \(\mathrm{38\%}\)
• This creates a range of plausible values: \(\mathrm{38\% ± 5.5\%}\)

3. Calculate the plausible range

• Lower bound: \(\mathrm{38\% - 5.5\% = 32.5\%}\)
• Upper bound: \(\mathrm{38\% + 5.5\% = 43.5\%}\)

4. APPLY CONSTRAINTS to select the correct interpretation

• We're looking for plausible values of the population percentage (not the sample percentage)
• We want a range (not a single value or direction)
• We want percentage (not number of students)

Answer: C. Plausible values of the percent of the students at the school who support a menu change are between \(\mathrm{32.5\%}\) and \(\mathrm{43.5\%}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students treat the sample statistic as the exact population value, not understanding that surveys estimate population parameters with uncertainty.

They think: "The survey found \(\mathrm{38\%}\), so \(\mathrm{38\%}\) of all students support the change." They ignore the margin of error concept entirely.

This may lead them to select Choice A (\(\mathrm{38\%}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse "plausible values of the number of students" with "plausible values of the percentage."

They calculate around the sample size (300 ± some small number around 5-10) instead of working with percentages, misunderstanding what population parameter is being estimated.

This may lead them to select Choice D (between 295 and 305)

The Bottom Line:

This problem tests whether students understand that sample surveys estimate population parameters within a range of uncertainty, rather than giving exact values. The key insight is recognizing that margin of error creates a confidence interval around the sample statistic.