Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Suppose 200 trout are caught, tagged, and released in a lake's general population. Let \(T\) denote the number of tagged fish that are recaptured when a sample of \(n\) trout are caught at a later date. The validity of the markrecapture method for estimating the lake's total trout population is based on the assumption that \(T\) is directly proportional to \(n\). If 10 tagged trout are recovered from a sample of 300 , estimate the total trout population of the lake.

Step-by-Step Solution

Answer

The estimated total trout population in the lake is 6000.

Step 1: Understand the Mark-Recapture Method

The mark-recapture method is a statistical technique for estimating animal populations. Initially, a known number of individuals (200 trout in this case) are captured, marked, and then released back into the population. Later, a second sample (300 trout) is captured, and the number of marked individuals (10 trout) found in this sample is recorded.

Step 2: Set Up the Proportional Relationship

According to the method, the ratio of marked fish in the second sample is approximately equal to the initial ratio of marked fish to the total population. This gives us the proportion \( \frac{T}{n} = \frac{200}{N} \), where \(T = 10\), \(n = 300\), and \(N\) is the total population we want to estimate.

Step 3: Solve for the Total Population

To find the total trout population, \(N\), we equate the two ratios: \( \frac{10}{300} = \frac{200}{N} \). To solve for \(N\), cross-multiply to get \(10N = 200 \times 300\). Simplify this to find \(N\):

Step 4: Calculate the Total Population

Continuing from Step 3, we have \(10N = 60000\). Solving for \(N\) gives \(N = \frac{60000}{10} = 6000\). Therefore, the estimated total trout population in the lake is \(6000\).

Key Concepts

Proportional Relationship

The concept of a proportional relationship is essential in understanding the mark-recapture method. This relationship implies that two quantities maintain a constant ratio. In the context of the exercise, it means that the number of tagged fish in the recaptured sample (10 trout) is proportional to the initial number of tagged fish (200 trout) compared to the total population of fish in the lake. The mathematical expression of this relationship is given as:\[\frac{T}{n} = \frac{200}{N}\]Here, \(T\) represents the tagged fish in the sample, \(n\) is the number of fish in the sample, and \(N\) is the total trout population. This equation suggests that the proportion of tagged fish in the sample should match the proportion of all initially tagged fish to the total population.

Statistical Technique

The mark-recapture method is a widely used statistical technique employed in population estimation. It helps researchers estimate the total number of individuals in a population, whether it be fish, animals, or other organisms. By using random sampling and marking techniques, scientists can infer the size of a population based on the proportion of marked and unmarked individuals. This technique assesses animal populations by initially tagging a known set of individuals and later recapturing a sample to observe how many of the tagged individuals are found again. The ratio between the recaptured tagged individuals and the total captured in the second round provides data to estimate overall population size. The primary assumption behind this method is that the tagged individuals mix homogeneously back into the general population, and the likelihood of capturing any individual is equal.

Population Estimation

Population estimation is a critical application of the mark-recapture method. By setting up a proportional relationship between the tagged and recaptured individuals and the entire population, experts can estimate the population size. In this exercise, the formula \(\frac{10}{300} = \frac{200}{N}\) is used to estimate the total trout population in the lake. Here, \(10\) represents the number of tagged fish recaptured, \(300\) is the size of the second sample, while \(200\) is the initially tagged fish. To determine \(N\), the exercise involves solving this equation, which leads to a straightforward computation that provides an estimation of the population size when access to the whole population isn't possible.

Cross-Multiplication

Cross-multiplication is a handy algebraic technique used to solve equations involving proportions. In this context, it helps estimate the total population (\(N\)) of trout in the lake. Using the proportion \(\frac{10}{300} = \frac{200}{N}\), cross-multiplying involves swapping and multiplying the numerator of one fraction by the denominator of the other fraction, leading to: \[10N = 200 \times 300\]By cross-multiplying, we rearrange the elements into a solvable equation. This simple method allows for quick resolution of proportions, helping to find unknown quantities in population estimation.Finally, solving this gives:\[N = \frac{60000}{10} = 6000\]Thus, cross-multiplication demonstrates its efficiency in converting proportions to solve for an unknown, making it fundamental in this statistical exercise.