A certain forest is 253 acres. To estimate the number of trees in the forest, a ranger randomly selects 5...

1. TRANSLATE the problem information

• Given information:
  • Total forest area: 253 acres
  • Sample data: 5 parcels of 1 acre each
  • Tree counts in sample parcels: 51, 59, 45, 52, 73
  • Need: Best estimate for total trees in entire forest based on sample mean

2. INFER the estimation strategy

• The sample mean gives us the best estimate for trees per acre across the entire forest
• To estimate total trees: multiply the estimated trees per acre by the total number of acres
• This approach assumes the sampled parcels are representative of the entire forest

3. SIMPLIFY by calculating the sample mean

• \(\mathrm{Mean} = (51 + 59 + 45 + 52 + 73) \div 5\)
• \(\mathrm{Mean} = 280 \div 5 = 56\) trees per acre

4. TRANSLATE the scaling calculation

• \(\mathrm{Total\ estimated\ trees} = \mathrm{Trees\ per\ acre} \times \mathrm{Total\ acres}\)
• \(\mathrm{Total\ estimated\ trees} = 56 \times 253\)
• \(\mathrm{Total\ estimated\ trees} = 14,168\) (use calculator)

5. APPLY CONSTRAINTS to select the correct range

• Looking at answer choices, 14,168 falls between 13,500 and 14,500

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that the mean provides the best estimate for the entire population. Students might instead use the minimum (45), median (52), or maximum (73) number of trees per acre.

For example, using the minimum: \(45 \times 253 = 11,385\) trees
This may lead them to select Choice A (11,000 to 12,000)

Or using the median: \(52 \times 253 = 13,156\) trees
This may lead them to select Choice B (12,500 to 13,500)

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what "based on the mean of the sample" means in the context of estimation. Some students might try to use the mean in a different way or get confused about the scaling process.

This leads to confusion and guessing among the available ranges.

The Bottom Line:

This problem tests whether students understand that sample statistics (especially the mean) are used to estimate population parameters, and that scaling from sample to population requires proportional thinking.