One way of estimating wildlife population of interest is to draw a sample of the population, tag the animals, and then return them to the population. Then, at a later date, draw another sample at random from the same population and compare the results. An ecologist using this methodology captures, tags, and then returns 198 fish to a lake. Three months later the ecologist captures a sample of 135 of the same type of fish, of which 22 were tagged. What would be the ecologist's best estimate for the number of fish of that type that are in the lake?

Answer

To estimate the wildlife population of fish in the lake, we can use the capture-recapture method. Here’s a step-by-step solution to find the best estimate for the number of fish in the lake:

### Step 1: Define Variables
Let:
- \( N \) = total population of fish in the lake (unknown)
- \( M \) = number of fish initially captured and tagged = 198
- \( n \) = number of fish captured in the second sample = 135
- \( K \) = number of tagged fish found in the second sample = 22

### Step 2: Set Up Proportions
The proportion of tagged fish in the second sample should be equal to the proportion of tagged fish in the entire population. This can be expressed as:
\[ \frac{M}{N} = \frac{K}{n} \]

### Step 3: Substitute Known Values
Substituting the known values into the proportion gives:
\[ \frac{198}{N} = \frac{22}{135} \]

### Step 4: Cross-Multiply to Solve for \( N \)
Cross-multiplying to solve for \( N \):
\[ 198 \cdot 135 = 22 \cdot N \]

### Step 5: Calculate the Left Side
Calculating the left side:
\[ 198 \cdot 135 = 26730 \]

### Step 6: Solve for \( N \)
Now, we can solve for \( N \):
\[ N = \frac{26730}{22} \]

### Step 7: Perform the Division
Calculating the division:
\[ N = 1215 \]

### Conclusion
The best estimate for the number of fish of that type in the lake is:
\[ \boxed{1215} \]

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