A researcher conducts a study to estimate the average height of adult males in a certain region. Based on a...

1. TRANSLATE the problem information

• Given information:
  • Sample estimate of average height = 70.2 inches
  • Margin of error = 1.8 inches
  • Need to find what we can conclude about the true population average

2. INFER the approach

• A margin of error creates a confidence interval around our sample estimate
• This interval tells us the range of plausible values for the true population parameter
• We need to calculate this interval and see which answer choice matches

3. TRANSLATE the margin of error into mathematical form

• Confidence interval formula: \(\mathrm{[estimate - margin\ of\ error,\ estimate + margin\ of\ error]}\)
• Substituting our values: \(\mathrm{[70.2 - 1.8,\ 70.2 + 1.8]}\)
• Calculating: \(\mathrm{[68.4,\ 72.0]}\)

4. INFER what this interval means

• The true average height of adult males in this region is plausibly anywhere between 68.4 and 72.0 inches
• Values outside this range are not supported by our data

5. APPLY CONSTRAINTS to select the correct answer

• Check each choice against our interval \(\mathrm{[68.4,\ 72.0]}\):
  • (A) Between 68.4 and 72.0 inches ✓ (exactly matches our interval)
  • (B) Less than 68.4 inches ✗ (outside our interval)
  • (C) Exactly 70.2 inches ✗ (too precise - we only have an estimate)
  • (D) Greater than 72.0 inches ✗ (outside our interval)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't understand what a confidence interval represents and think the sample estimate (70.2) is the exact population value.

They see "average height is 70.2 inches" and conclude this must be exactly right, ignoring the margin of error entirely. This may lead them to select Choice C (exactly 70.2 inches).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "margin of error" means mathematically and don't know how to create the confidence interval.

They might think margin of error means "plus or minus from any value" rather than "plus or minus from the sample estimate." This leads to confusion about which values are actually plausible, causing them to get stuck and guess randomly.

The Bottom Line:

This problem tests whether students understand that statistical estimates come with uncertainty. The key insight is that a margin of error creates a range of plausible values, not a single exact answer.