A factory produces 80,000 electronic components each month. A quality control team randomly selected and tested 1,200 components from the...

1. TRANSLATE the problem information

• Given information:
  • Monthly production: 80,000 components
  • Sample tested: 1,200 components
  • Estimated defect rate: 4.2%
  • Margin of error: ±0.5%

• What this tells us: We need to find a plausible range, not an exact value

2. INFER the statistical concept

• Key insight: Margin of error means our 4.2% estimate could be off by 0.5% in either direction
• This creates a range of possible defect rates:
  • Lowest possible rate: \(4.2\% - 0.5\% = 3.7\%\)
  • Highest possible rate: \(4.2\% + 0.5\% = 4.7\%\)

3. TRANSLATE percentages to decimals and calculate range

• Convert to decimal form for calculation:
  • Lower bound: \(3.7\% = 0.037\)
  • Upper bound: \(4.7\% = 0.047\)

• Apply to total production of 80,000:
  • Minimum defectives: \(80,000 \times 0.037 = 2,960\)
  • Maximum defectives: \(80,000 \times 0.047 = 3,760\)

4. APPLY CONSTRAINTS to select the answer

• The plausible range is \([2,960, 3,760]\) defective components
• Check each answer choice:
  • (A) 504: Too low (below 2,960)
  • (B) 2,240: Too low (below 2,960)
  • (C) 3,120: ✓ Within range
  • (D) 3,920: Too high (above 3,760)

Answer: C (3,120)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students focus only on the 4.2% estimate and ignore the margin of error completely.

They calculate: \(80,000 \times 0.042 = 3,360\) defective components, then look for the closest answer choice. Since 3,360 isn't an option, they might select Choice C (3,120) by proximity, getting the right answer for wrong reasons, or select Choice D (3,920) as another "close" option.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret margin of error as applying to the final count rather than the percentage rate.

They calculate 4.2% of 80,000 = 3,360, then add/subtract 0.5% of 80,000 (400), giving them a range of [2,960, 3,760]. This accidentally leads to the correct range, but the reasoning process was flawed - they applied the margin of error to the wrong quantity.

The Bottom Line:

This problem tests understanding that statistical estimates come with uncertainty ranges, not precise values. Success requires recognizing that margin of error modifies the rate itself, creating a range of plausible outcomes rather than a single "best guess."