A school band has 140 members. In a random sample of 20 band members, 5 said they plan to attend...

1. TRANSLATE the problem information

• Given information:
  • Total band members: 140
  • Sample surveyed: 20 members
  • In sample, 5 plan to attend workshop
• Find: Estimate of total members planning to attend

2. INFER the approach

• Key insight: The proportion planning to attend in our sample should be similar to the proportion in the entire band
• Strategy: Find the sample proportion, then apply it to the total population
• Two equivalent methods can work here

3. Calculate the sample proportion

• Sample proportion = \(\mathrm{5 ÷ 20}\) = \(\mathrm{\frac{1}{4}}\) = \(\mathrm{0.25}\) = \(\mathrm{25\%}\)

4. SIMPLIFY using either scaling method

Method A (Scaling factor):

• Total-to-sample ratio: \(\mathrm{140 ÷ 20}\) = \(\mathrm{7}\)
• Expected total: \(\mathrm{5 × 7}\) = \(\mathrm{35}\)

Method B (Percentage):

• Apply 25% to total population: \(\mathrm{0.25 × 140}\) = \(\mathrm{35}\)

Answer: D. 35

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to scale the sample result to the full population. They see "5 said they plan to attend" and think the answer is simply 5, not understanding that this represents only the sample group.

This leads them to select Choice A (5).

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly recognize they need to do something with the numbers but misinterpret what the question is asking. They might think "20 people were asked, so 20 people plan to attend" or get confused about which numbers to use in their calculation.

This may lead them to select Choice B (20) or causes them to make calculation errors that result in Choice C (25).

The Bottom Line:

The key challenge is recognizing that sample data must be scaled up to represent the whole population - students need to move from "5 out of 20 in my sample" to "how many out of 140 in total?"