In a production batch of 25{,000} smartphones, 500 were randomly selected for quality control testing. The testing found that an...

1. TRANSLATE the statistical information

• Given information:
  • Total batch: 25,000 smartphones
  • Sample tested: 500 smartphones
  • Estimated defective rate: \(8\% \pm 2\%\) margin of error
  • Need: Total number with NON-defective batteries

2. INFER the approach strategy

• The margin of error means the true defective rate could be 2 percentage points higher or lower than 8%
• Since we want non-defective smartphones, we need the complement of whatever defective range we calculate
• We'll apply this range to the full population of 25,000

3. SIMPLIFY to find the defective rate range

• Lower bound for defective rate: \(8\% - 2\% = 6\%\)
• Upper bound for defective rate: \(8\% + 2\% = 10\%\)

4. INFER the complement for non-defective rates

• If 10% are defective → \(100\% - 10\% = 90\%\) are non-defective
• If 6% are defective → \(100\% - 6\% = 94\%\) are non-defective
• Non-defective range: 90% to 94%

5. SIMPLIFY to calculate total non-defective smartphones

• Lower bound: \(90\% \times 25,000 = 22,500\) smartphones
• Upper bound: \(94\% \times 25,000 = 23,500\) smartphones
• Plausible range: \([22,500, 23,500]\)

6. APPLY CONSTRAINTS to select the correct answer

• Check which answer choice falls within \([22,500, 23,500]\):
  • A. 460 (way too small)
  • B. 2,500 (too small)
  • C. 23,150 ✓ (fits perfectly in range)
  • D. 24,500 (exceeds upper bound)

Answer: C. 23,150

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students calculate the number of defective smartphones instead of non-defective ones, missing the word "non-defective" in the question.

They correctly find the 6%-10% defective range, then calculate:

• Lower bound: \(6\% \times 25,000 = 1,500\) defective
• Upper bound: \(10\% \times 25,000 = 2,500\) defective

Since 2,500 appears as choice B, this may lead them to select Choice B (2,500).

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students calculate the correct range \([22,500, 23,500]\) but don't systematically check which answer choice falls within these bounds.

They might see that 24,500 is "close to" their upper bound of 23,500 and think it's acceptable, leading them to select Choice D (24,500).

The Bottom Line:

This problem tests whether students can work with statistical ranges AND carefully track what quantity they're actually calculating (defective vs. non-defective). The key insight is that margin of error creates a range, and we must apply that range to find plausible values for the complement.