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
R/n = m/n,
and hence that the estimated population size is
N = (M × 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 €œtagged€ by having returned their census form and being recorded as counted) and the recapture phase (the second sample) is the post enumerative survey (PES) conducted after the census. Label the table in terms of the census application.

In First Sample? Yes (tagged No (not tagged) Total In Second Yes In Sample? No Total