A wildlife biologist samples 800 fish from a lake with an estimated population of 40{,}000 fish. From the sample, it...

1. TRANSLATE the statistical information

• Given information:
  • Total population: 40,000 fish
  • Sample estimate: 62% are at least two years old
  • Margin of error: 3 percentage points
  • Find: number of fish LESS than two years old

• The margin of error means the true percentage could be 3% above or below 62%

2. INFER the range for 'at least two years old'

• With margin of error, the percentage at least two years old could be:
  • Lowest: \(62\% - 3\% = 59\%\)
  • Highest: \(62\% + 3\% = 65\%\)

• So between 59% and 65% of fish are at least two years old

3. INFER the complementary range for 'less than two years old'

• Since fish are either 'at least two years old' OR 'less than two years old':
  • If 65% are at least two years old → \(100\% - 65\% = 35\%\) are less than two years old
  • If 59% are at least two years old → \(100\% - 59\% = 41\%\) are less than two years old

• So between 35% and 41% of fish are less than two years old

4. Calculate the population range

• Minimum fish less than two years old: \(35\% \times 40,000 = 14,000\)
• Maximum fish less than two years old: \(41\% \times 40,000 = 16,400\)

• Plausible range: 14,000 to 16,400 fish

5. APPLY CONSTRAINTS to select the answer

• Check which choice falls within [14,000, 16,400]:
  • (A) 12,000 - too low
  • (B) 13,800 - too low
  • (C) 15,200 - ✓ within range
  • (D) 17,600 - too high

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret the margin of error, thinking it only applies in one direction or confusing it with the sample size uncertainty.

Some students might calculate \(62\% \pm 3\%\) but then apply this directly to find fish less than two years old, getting something like \(38\% \pm 3\%\), which gives 35% to 41% - the right range but through flawed reasoning. Others might ignore the margin of error entirely and just calculate \(38\% \times 40,000 = 15,200\), coincidentally getting the right answer.

This coincidental match might lead them to select Choice C (15,200) for the wrong reasons, masking their conceptual confusion.

Second Most Common Error:

Poor INFER skill: Students correctly find that 59% to 65% are at least two years old but fail to recognize that the remaining fish represent the 'less than two years old' category.

Instead, they might try to directly apply the \(62\% \pm 3\%\) to the 'less than two years old' group, thinking this gives \(62\% \pm 3\% = 59\%\) to 65% are less than two years old. This leads to \(59\% \times 40,000 = 23,600\) to \(65\% \times 40,000 = 26,000\), which doesn't match any answer choice exactly but is closest to Choice E (24,800).

The Bottom Line:

This problem requires careful interpretation of what margin of error means and systematic application of complementary percentages. Students who rush through the percentage calculations or misunderstand the relationship between the two age categories will struggle to identify the correct range.