In the experiment of Example 7.1.8, suppose 200 animals are...

In the experiment of Example 7.1.8, suppose 200 animals are tagged, 20 are captured or recaptured, and 4 of the 20 are discovered to be tagged so that the maximum likelihood estimate of the population size is $$ \widehat{N}=\left[\frac{200 \times 20}{4}\right]=1,000 \text { animals. } $$ (a) If the actual value of $N$ is 503 , calculate the probability that 4 or fewer tagged animals are found in a sample of 20 captured or recaptured animals. Calculate the binomial approximation as well. (b) If $N=2,790$, what is the probability that 4 or more tagged animals are found in a sample of 20 recaptured animals? What is the binomial approximation of this value? 18. [C5]

In the experiment of Example 7.1.8, suppose 200 animals are tagged, 20 are captured or recaptured, and 4 of the 20 are discovered to be tagged so that the maximum likelihood estimate of the population size is
$$
\widehat{N}=\left[\frac{200 \times 20}{4}\right]=1,000 \text { animals. }
$$
(a) If the actual value of $N$ is 503 , calculate the probability that 4 or fewer tagged animals are found in a sample of 20 captured or recaptured animals. Calculate the binomial approximation as well.
(b) If $N=2,790$, what is the probability that 4 or more tagged animals are found in a sample of 20 recaptured animals? What is the binomial approximation of this value?
18. [C5]

Key Concepts

Hypergeometric Distribution

The hypergeometric distribution models the probability of drawing a specific number of successes (for example, tagged animals) from a finite population without replacement. It is particularly applicable in situations where the sample size is a significant portion of the whole population, leading to probabilities that change as each draw is made. In mark?recapture studies, the hypergeometric distribution provides a framework for calculating the likelihood of capturing a certain number of tagged individuals given the total number of tagged animals and the total population size.

Binomial Approximation

The binomial distribution approximates the count of successes in a fixed number of independent trials, each with the same probability of success. When the population is large and the sample size is relatively small compared to it, the hypergeometric distribution can be approximated by the binomial distribution, simplifying calculations. This approximation assumes that each draw is nearly independent, which is a reasonable assumption when the removal of a sampled individual has a negligible effect on the overall probability of success in subsequent draws.

Mark-Recapture Method

The mark-recapture method is a field technique used in ecology to estimate the size of wildlife populations. By capturing individuals, marking or tagging them, and then recapturing individuals in a subsequent sample, researchers use the proportion of marked individuals found in the second sample to infer the total population size. This technique assumes that the marked individuals have mixed thoroughly with the rest of the population and that capture probabilities are equal for all individuals.

Maximum Likelihood Estimation (MLE)

Maximum likelihood estimation is a statistical method used to estimate the parameters of a model by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. In the context of mark?recapture, MLE is used to infer the total population size based on the observed numbers of marked and recaptured individuals, translating the problem into an estimation issue where the parameters are assessed for their likelihood given the sampling data.

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