A quality control manager randomly selected 20 products from a production batch of 500 products. She found that 4 of...

1. TRANSLATE the problem information

• Given information:
  • Total batch size: 500 products
  • Sample size: 20 products (randomly selected)
  • Defective products in sample: 4 products
  • Need to estimate defective products in entire batch

2. INFER the approach

• The key insight is that if the sample was randomly selected, the proportion of defective products in the sample should be similar to the proportion in the entire batch
• Strategy: Calculate sample proportion, then apply it to the whole batch

3. SIMPLIFY to find the sample proportion

• Sample proportion = 4 defective ÷ 20 total = \(\frac{4}{20} = \frac{1}{5} = 0.2\)
• This means 20% of the sample had defects

4. INFER how to estimate for the entire batch

• If 20% of the sample had defects, we estimate that 20% of the entire batch has defects
• Apply the proportion: \(0.2 \times 500 = 100\)

Answer: D (100)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skills: Students don't recognize that they need to scale up from sample to population using proportions. Instead, they might think the question is asking for something directly given in the problem.

This leads them to select Choice A (4) - the actual number of defective products found in the sample, or Choice B (20) - the sample size itself.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the proportion concept but make arithmetic errors when calculating \(\frac{4}{20}\) or when multiplying by 500.

This may lead to confusion and guessing among the remaining choices, or potentially selecting Choice C (80) if they incorrectly calculate the proportion.

The Bottom Line:

This problem tests whether students understand that random sampling allows us to make estimates about larger populations using proportions, not just report the raw numbers from the sample.