A worker at a local aquarium is instructed to estimate the population of anchovies in the tank. She catches 32 anchovies, marks them with special paint, and returns them to the tank. Then she catches 950 anchovies and finds that 16 of them are marked. To the nearest whole number, what is the best estimate for the anchovy population?

To estimate the total population of anchovies in the tank, we can again use the **capture-recapture method**.

Let \( N \) be the total anchovy population. The proportion of marked anchovies in the second sample should equal the proportion of marked anchovies in the entire population.

Given:
- Number of anchovies initially marked (\( M \)) = 32
- Number of marked anchovies in the second sample (\( m \)) = 16
- Total number of anchovies in the second sample (\( n \)) = 950

The proportion can be set up as:

\[
\frac{16}{950} = \frac{32}{N}
\]

To solve for \( N \), cross-multiply:

\[
16N = 32 \times 950
\]

\[
N = \frac{32 \times 950}{16}
\]

Now calculate \( N \):

\[
N = \frac{32 \times 950}{16} = \frac{30400}{16} = 1900
\]

So, the best estimate for the anchovy population in the tank is approximately **1,900**.

Would you like more details or have any questions?

Here are some related questions to consider:
1. How does the number of marked individuals in the second sample affect the population estimate?
2. What factors could lead to inaccuracies in the capture-recapture method?
3. How can researchers ensure that the marking process does not affect the behavior of the animals?
4. What are some alternative methods for estimating animal populations in controlled environments?
5. How could the aquarium worker improve the accuracy of her population estimate?

**Tip:** When performing population estimates, it's crucial to ensure that the marked individuals have sufficient time to mix evenly with the rest of the population before the second sample is taken.

Learn how to estimate the population of anchovies in a tank using the capture-recapture method. This problem involves statistical analysis and probability calculations based on marked and unmarked samples. Suitable for Grades 10-12 students studying statistics and probability.

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