A factory produced a batch of 480 smartphones. For quality control, a random sample of 80 phones was selected from...

1. TRANSLATE the problem information

• Given information:
  • Total batch: 480 smartphones
  • Sample size: 80 phones
  • Defective phones in sample: 12
  • Need: Total defects in entire batch

2. INFER the approach

• This is a proportion problem using sampling
• The key insight: If 12 out of 80 sample phones are defective, then the same proportion should apply to the entire batch
• We need to find what fraction of the sample is defective, then apply that fraction to the total batch

3. SIMPLIFY the sample defect rate

• Defect rate = \(\frac{12}{80} = \frac{3}{20} = 0.15 = 15\%\)
• This means 15% of phones in our sample are defective

4. INFER the final calculation needed

• If 15% of the sample is defective, then approximately 15% of the entire batch should be defective
• Apply this rate to the total: \(15\%\) of 480 phones

5. SIMPLIFY the final calculation

• \(0.15 \times 480 = 72\) (use calculator)
• Alternative: \(\frac{3}{20} \times 480 = 3 \times 24 = 72\)

Answer: B (72)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing this as a proportion problem requiring the sample rate to be applied to the entire population.

Students might think the 12 defective phones found IS the total number of defects, failing to understand that the sample represents the whole batch. This leads them to select Choice A (12).

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what quantity needs to be calculated or confusing sample size with the answer.

Students might think since 80 phones were tested, that's somehow the answer, leading them to select Choice C (80). Others might calculate \(480 - 72 = 408\), thinking about non-defective phones instead of defective ones, selecting Choice D (408).

The Bottom Line:

This problem tests whether students understand that sample statistics can estimate population parameters through proportional reasoning. The key insight is recognizing that "\(\frac{12}{80}\)" creates a rate that applies to the entire batch.