A process called tag-and-recapture is used to estimate...

A process called tag-and-recapture is used to estimate populations of animals in the wild. First, some number $T$ of the animals are captured, tagged and released into the wild. Later, a number $S$ of the animals are captured, of which $t$ are observed to be tagged. The estimate of the total population is then $P(T, S, t)=\frac{T S}{t} .$ Compute $P(100,60,15) ;$ the proportion of tagged animals in the recapture is $\frac{15}{60}=\frac{1}{4}$. Based on your estimate of the total population, what proportion of the total population has been tagged? Now compute $\frac{\partial P}{\partial t}(100,60,15)$ and use it to estimate how much your population estimate would change if one more recaptured animal were tagged.

Key Concepts

Tag-and-Recapture Method

This is a statistical technique used in ecology to estimate an animal population's size by capturing a number of individuals, marking them, releasing them, and then recapturing a sample after the marked individuals have mixed back into the population. The proportion of marked individuals in the second sample is then used to approximate the population total.

Proportional Estimation

This concept refers to the use of ratios from samples to infer properties of a whole population. In the tag-and-recapture method, the ratio of tagged individuals in the recapture sample is equated to the ratio of initially tagged individuals in the entire population to derive an estimate of the total population.

Partial Derivative

A partial derivative measures how a function changes as one of its variables changes, while all other variables are held constant. In this context, computing the partial derivative with respect to the number of recaptured tagged animals helps to understand how sensitive the population estimate is to changes in that specific variable.

Sensitivity Analysis

This is a technique used to determine how the variation in the output of a model can be attributed to different variations in the input. By using the partial derivative in the estimation, one can approximate how a small increase in the number of captured tagged animals would affect the overall population estimation.

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