A city has 50 city council members. A reporter polled a random sample of 20 city council members and found...

1. TRANSLATE the problem information

• Given information:
  • Total city council members: 50
  • Random sample size: 20 members
  • Members in sample supporting bill: 6
• What we need: Best estimate of total members who support the bill

2. INFER the approach

• Since this is a random sample, the proportion of supporters in the sample should be approximately equal to the proportion of supporters in the entire population
• Strategy: Find sample proportion, then apply it to total population

3. SIMPLIFY to find the sample proportion

• Proportion supporting in sample = \(\frac{6}{20} = 0.3 = 30\%\)

4. SIMPLIFY to apply proportion to total population

• Estimated total supporters = \(50 \times 0.3 = 15\) members

Answer: C. 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize that they need to scale up from the sample to the population. They see "6 supported the bill" and think that's the final answer without considering that this was only from a sample of 20 out of 50 total members.

This leads them to select Choice A (6).

Second Most Common Error:

Poor TRANSLATE execution: Students misinterpret what the numbers represent. They might think "20 were polled, so 30 weren't polled" and confuse this with the answer, or they make calculation errors when finding the proportion.

This may lead them to select Choice D (30) or Choice B (9).

The Bottom Line:

This problem tests whether students understand the fundamental principle of statistical sampling: that a random sample should be representative of the larger population, so proportions found in the sample can be used to estimate proportions in the population.