A sample of 40 fourth-grade students was selected at random from a certain school. The 40 students completed a survey...

1. TRANSLATE the problem information

• Given information:
  • Random sample: 40 fourth-grade students from a certain school
  • Survey topic: Morning announcements
  • Result: 32 found announcements helpful
  • Question: Largest population for applying results

2. INFER the sampling relationship

• Key insight: Random sampling allows us to generalize results, but only back to the population from which we sampled
• The original population sampled = "fourth-grade students at a certain school"
• We cannot extend beyond this population

3. APPLY CONSTRAINTS to evaluate each choice

• Choice A: Only the 40 students surveyed - This is just describing the sample, not generalizing
• Choice B: All fourth-grade students at the school - This matches our original population exactly
• Choice C: All students at the school - This includes other grades that weren't represented in our sample
• Choice D: All fourth-grade students in the county - This goes far beyond our school's fourth-graders

4. Select the largest valid population

• Among valid options, Choice B represents the largest population we can generalize to

Answer: B. All fourth-grade students at the school

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students think "bigger is better" and choose the largest-sounding population without considering sampling limitations.

They reason: "If we learned something about fourth-graders, it must apply to all fourth-graders everywhere" or "The more people we can apply this to, the better the answer."

This may lead them to select Choice D (All fourth-grade students in the county)

Second Most Common Error:

Inadequate TRANSLATE reasoning: Students misunderstand what "largest population" means in the context of statistical validity.

They interpret the question as asking for the biggest group mentioned rather than the biggest group to which results can be validly applied. This leads them to focus on scope rather than statistical legitimacy.

This may lead them to select Choice C (All students at the school)

The Bottom Line:

This problem tests understanding that statistical generalization has boundaries - you can only apply sample results to the population you actually sampled from, regardless of how much you'd like to extend those conclusions further.