Question:To estimate the number of students in a school of 800 students who prefer a certain subject, a random sample...

1. TRANSLATE the problem information

• Given information:
  • Total students in school: 800
  • Sample proportion preferring the subject: 0.35
  • Margin of error: 0.06

• We need to find the plausible range for the actual number of students who prefer the subject.

2. INFER the approach

• The margin of error tells us there's uncertainty in our estimate
• This creates a confidence interval: estimate ± margin of error
• We need to convert this proportion-based interval into actual student counts

3. TRANSLATE the confidence interval to actual students

• Confidence interval for proportion: \(\mathrm{0.35 \pm 0.06}\)
• Lower bound: \(\mathrm{0.35 - 0.06 = 0.29}\)
• Upper bound: \(\mathrm{0.35 + 0.06 = 0.41}\)

4. TRANSLATE proportions to student counts

• Lower bound: \(\mathrm{0.29 \times 800 = 232}\) students
• Upper bound: \(\mathrm{0.41 \times 800 = 328}\) students

5. INFER the meaning for answer selection

• The confidence interval means it's plausible the true number is anywhere from 232 to 328 students
• We cannot say the number is exactly 280 (the point estimate)
• We cannot say it's definitely outside this range

Answer: A (It is plausible that the number is between 232 and 328)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Misunderstanding what confidence intervals represent

Students often think the point estimate (\(\mathrm{0.35 \times 800 = 280}\)) gives the exact answer, ignoring the margin of error entirely. They see "0.35" as the definitive proportion and conclude exactly 280 students prefer the subject.

This may lead them to select Choice C (The number is exactly 280)

Second Most Common Error:

Incomplete TRANSLATE execution: Only calculating the point estimate

Students correctly multiply \(\mathrm{0.35 \times 800 = 280}\) but fail to incorporate the margin of error into their calculation. They recognize there's uncertainty but don't know how to work with the \(\mathrm{\pm 0.06}\) to find the full range.

This leads to confusion and guessing between the remaining choices.

The Bottom Line:

Confidence intervals are about ranges of plausible values, not exact predictions. The key insight is that margin of error must be converted to actual student counts (\(\mathrm{0.06 \times 800 = 48}\)) to create the meaningful range of 232 to 328 students.