At a high school with 1,200 students, 42% are juniors. A counselor randomly surveys 25 juniors and finds that 18...

1. TRANSLATE the problem information

• Given information:
  • Total students: 1,200
  • 42% are juniors
  • Sample: 25 juniors surveyed
  • Sample result: 18 plan to attend workshop
• We need to estimate total juniors who will attend

2. INFER the solution strategy

• The key insight: Use the sample proportion to estimate what proportion of ALL juniors will attend
• Strategy: Find total juniors → Find sample proportion → Apply proportion to all juniors

3. SIMPLIFY to find total number of juniors

• Calculate: \(42\% \text{ of } 1,200 = 0.42 \times 1,200 = 504\) juniors

4. TRANSLATE the sample data into a proportion

• Sample proportion attending: \(\frac{18}{25} = 0.72\) (or 72%)

5. SIMPLIFY to find the final estimate

• Apply this proportion to all juniors: \(0.72 \times 504 = 362.88\)
• Round to nearest whole person: 363

Answer: B) 363

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students apply the sample proportion to the wrong population group.

Instead of applying \(\frac{18}{25}\) to the 504 juniors, they apply it to all 1,200 students: \(\frac{18}{25} \times 1,200 = 864\). This completely misses that the survey only tells us about junior behavior, not all student behavior.

This leads them to select Choice E (864).

Second Most Common Error:

Poor TRANSLATE reasoning: Students use the raw sample numbers instead of the proportion.

They might calculate: 504 juniors total, 18 out of 25 in sample attend, so they think \(504 - 25 + 18 = 497\) or make some other incorrect arithmetic with the raw numbers. This leads to confusion and guessing among the middle answer choices.

The Bottom Line:

This problem tests whether students understand that sample proportions estimate population proportions, and that the population being estimated must match the population that was sampled. The junior sample tells us about juniors, not about all students.