Scott selected 20 employees at random from all 400 employees at a company. He found that 16 of the employees...

1. TRANSLATE the problem information

• Given information:
  • Total employees at company: 400
  • Sample size: 20 employees (selected randomly)
  • Sample result: 16 out of 20 are enrolled in exactly three courses
• What we need to find: Number of all 400 employees enrolled in exactly three courses

2. INFER the approach

• Since this is a random sample, we can assume the sample proportion represents the population proportion
• Key insight: If 16 out of 20 sample employees have this characteristic, then the same proportion of all 400 employees should have it
• Strategy: Find the sample proportion, then apply it to the total population

3. SIMPLIFY the sample proportion

• Sample proportion = \(\frac{16}{20} = 0.8 = 80\%\)
• This means 80% of the sample employees are enrolled in exactly three courses

4. INFER the population estimate

• If 80% of the sample has this characteristic, our best estimate is that 80% of all employees have it
• Population estimate = \(80\% \times 400\) employees

5. SIMPLIFY the final calculation

• \(80\% \times 400 = 0.8 \times 400 = 320\) employees

Answer: B. 320

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what quantity they're solving for and focus on the wrong numbers from the problem.

They might calculate \(20 - 16 = 4\) (employees in sample NOT enrolled in three courses) or \(400 - 20 = 380\) (employees not selected for the sample), thinking these represent the answer to the question.

This may lead them to select Choice A (4) or Choice C (380).

Second Most Common Error:

Poor INFER reasoning: Students don't recognize that this requires proportional thinking and instead try to use the sample numbers directly.

They might think the answer is simply \(16\) (the number from the sample) or make calculation errors by setting up incorrect relationships, leading to confusion about how sample data relates to population estimates.

This may lead them to select Choice D (384) or causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students understand that random samples can be used to make population estimates through proportional reasoning. The key insight is recognizing that sample percentages serve as our best estimate for population percentages.