Problem Set 5 - MS&E 226 Small Data Problem Set 5 Due...

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MS&E 226 Problem Set 5 “Small” Data Due: December 1, 2016, 5:00 PM , submitted through Gradescope Problem 1. Often, biologists are interested in estimating the size of an animal’s population size. Frequently it is impossible to count all animals, so we have to resort to estimation. A popular method is called capture-recapture. First, a portion of the population is captured, marked and released. At a later time, another portion is captured and the number of marked individuals in this second sample is counted. We assume that all animals have the same probability of being captured is identical. Let N be the total number of animals in the population, and let n denote the number of animals marked on the first visit. Furthermore, let K be the number of animals captured on the second visit and k be the number of marked recaptured animals. For this problem, we assume that in both samples the animals are sampled without replacement. A estimator of the total population is the Lincoln-Peterson estimator . We expect that the pro- portion of marked individuals ( k/K ) should be approximately equal to the total population that is marked ( n/N ). Hence, the estimator then becomes: ˆ N LP = Kn k . However, it turns out to be biased in small samples. Instead, we consider the Chapman estima- tor : ˆ N C = ( K + 1)( n + 1) k + 1 - 1 .
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