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Capture-recapture Biologists and naturalists often use sampling to estimate sizes of populations, such as deer or fish, for which a census is impossible. Capture-recapture is one method for doing this. A biologist wants to count the deer population in a certain region. She captures 50 deer, tags each, and then releases them. Several weeks later, she captures 125 deer and finds that 12 of them were tagged. Let \(N=\) population size, \(M=\) size of first sample, \(n=\) size of second sample, \(R=\) number tagged in second sample. The table shows how results can be summarized. a. Identify the values of \(M, n,\) and \(R\) for the biologist's experiment. b. One way to estimate \(N\) lets the sample proportion of tagged deer equal the population proportion of tagged deer. Explain why this means that $$ \frac{R}{n}=\frac{M}{N} $$ and hence that the estimated population size is \(N=(M \times n) / R\) c. Estimate the number of deer in the deer population using the numbers given. d. The U.S. Census Bureau uses capture-recapture to make adjustments to the census by estimating the undercount. The capture phase is the census itself (persons are "tagged" by having returned their census form and being recorded as counted) and the recapture phase (the second sample) is the postenumerative survey (PES) conducted after the census. Label the table in terms of the census application.

Step by step solution

01

Identify Sample Values

For the biologist's experiment, identify the values given: - First sample size, or marked deer, is denoted as \(M = 50\).- Second sample size, total deer captured in the second attempt, is \(n = 125\).- Number of marked deer recaptured, \(R = 12\). These values set the framework for estimating the deer population.

02

Understand Proportions in Sampling

The equation for the capture-recapture method is based on the premise that the proportion of tagged animals in the second sample should represent the proportion of tagged animals in the entire population. Thus: \[ \frac{R}{n} = \frac{M}{N} \]This equation indicates that the fraction of tagged deer in the second sample equals the fraction of tagged deer in the population. From this proportion, we can derive a formula to estimate the population size \(N\).

03

Derive the Population Size Formula

Starting from the equation \(\frac{R}{n} = \frac{M}{N}\), solve for \(N\) by cross-multiplying to get: \[ R \times N = M \times n \]Dividing both sides by \(R\) results in the equation to estimate \(N\):\[ N = \frac{M \times n}{R} \]

04

Calculate the Deer Population Estimate

With the values \(M = 50\), \(n = 125\), and \(R = 12\), substitute these into the formula:\[ N = \frac{50 \times 125}{12} \]Calculate the result:\[ N = \frac{6250}{12} \approx 520.83 \]Rounding yields an estimated deer population of about 521.

05

Relate to Census Application

In the context of the U.S. Census Bureau, the table values represent:- \(M\): Number of people counted in the actual census (participants who returned their forms).- \(n\): Total number of people in the postenumerative survey (PES).- \(R\): Number of individuals from the PES that were also recorded in the census (recaptured in PES). This analogy helps estimate population adjustments due to undercounting.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Estimation

Understanding how biologists estimate population sizes can initially seem like solving a complex puzzle. However, with the capture-recapture method, it becomes much more manageable. Here's how it works: Imagine you're trying to determine the number of deer in a forest. You can't count each deer individually, so how would you do it? Well, this method comes in handy.

The core idea is to link the proportion of a tagged sample to the entire population. Biologists first capture a sample of deer, tag them, and let them back into the wild. Later, another sample of deer is captured, and the number of tagged deer within this group is noted. This process is known as capture-recapture and forms the basis for population estimation without directly counting every single individual. By using proportions, the biologist can estimate the total population size based on the idea that the proportion of tagged to untagged animals in the sample represents that of the whole population.

For example, if 12 out of 125 captured deer were previously tagged, this ratio helps biologists extrapolate the estimated total population using the formula: \[N = \frac{M \times n}{R}\] Substituting the known values yields an estimated population size.

Sampling Methods

Sampling methods are a game-changer when dealing with large populations. Imagine trying to count all the fish in a lake; it's practically impossible. Instead, biologists rely on sampling to make informed guesses. The capture-recapture is a specific method of sampling that plays a vital role in ecological studies.

There are two main sampling events in this method: the initial capture and tagging phase, and the subsequent recapture phase. Let's break it down further:

• The first capture, where a portion of the population is captured, tagged, and released back into their habitat. This group provides an initial snapshot of the population.
• The recapture, which occurs after an appropriate time has elapsed, involves capturing a new sample and checking for previously tagged individuals. This helps infer the proportion of the overall population.

This sampling method's efficiency relies upon the assumption that tagged individuals have had enough time to mix uniformly with the rest of the population, thus providing reliable data for population estimates. It's a clever way to use small, manageable samples to understand a much larger picture.

Data Analysis

Data analysis in the capture-recapture method involves understanding and leveraging proportions to make global predictions. In our deer example, the analysis revolves around a critical assumption: the proportion of tagged deer in the second sample represents the proportion in the overall population.

By applying the formula \(\frac{R}{n} = \frac{M}{N}\), biologists link sample statistics to population parameters. Here's the breakdown:

• Proportion of tagged deer in the recapture (\(R/n\)): This tells us how common tagged deer are in the recaptured sample.
• Proportion of tagged deer in the entire population (\(M/N\)): This predicts how common tagged deer should be across the whole group.

By equating these proportions, we derive \(N\), an estimated population size. The analysis assumes a random mix has occurred, no tags lost, and that the population size remains constant during the sampling process.

An error margin always exists due to potential biases like behavior changes in tagged animals, but despite this, the capture-recapture method remains a robust tool for ecological population estimates.