2. Excursions in Modern Mathematics, 7e: 13.5 - 2Copyright © 2010 Pearson Education, Inc. 13 Collecting Statistical Data 13.1The Population 13.2Sampling 13.3 Random Sampling 13.4Sampling: Terminology and Key Concepts 13.5The Capture-Recapture Method 13.6Clinical Studies

3. Excursions in Modern Mathematics, 7e: 13.5 - 3Copyright © 2010 Pearson Education, Inc. We have already observed that finding the exact N-value of a large and elusive population can be extremely difficult and sometimes impossible. In many cases, a good estimate is all we really need, and such estimates are possible through sampling methods. The simplest sampling method for estimating the N-value of a population is called the capture-recapture method. Finding the N-value

4. Excursions in Modern Mathematics, 7e: 13.5 - 4Copyright © 2010 Pearson Education, Inc. ■ Step 1. Capture (sample): Capture (choose) a sample of size n 1, tag (mark, identify) the animals (objects, people), and release them back into the general population. THE CAPTURE-RECAPTURE METHOD

5. Excursions in Modern Mathematics, 7e: 13.5 - 5Copyright © 2010 Pearson Education, Inc. ■ Step 2. Recapture (resample): After a certain period of time, capture a new sample of size n 2 and take an exact head count of the tagged individuals (i.e., those that were also in the first sample). Call this number k. THE CAPTURE-RECAPTURE METHOD

6. Excursions in Modern Mathematics, 7e: 13.5 - 6Copyright © 2010 Pearson Education, Inc. ■ Step 3. Estimate: The N-value of the population can be estimated to be approximately (n 1 n 2 )/k. THE CAPTURE-RECAPTURE METHOD

7. Excursions in Modern Mathematics, 7e: 13.5 - 7Copyright © 2010 Pearson Education, Inc. The capture-recapture method is based on the assumption that both the captured and recaptured samples are representative of the entire population. Under these assumptions, the proportion of tagged individuals in the recaptured sample is approximately equal to the proportion of the tagged individuals in the population. In other words, the ratio k/n 2 is approximately equal to the ratio n 1 /N. From this we can solve for N and get N ≈ (n 1 n 2 )/k. Capture-Recapture Method

8. Excursions in Modern Mathematics, 7e: 13.5 - 8Copyright © 2010 Pearson Education, Inc. A large pond is stocked with catfish. As part of a research project we need to estimate the number of catfish in the pond. An actual head count is out of the question (short of draining the pond), so our best bet is the capture- recapture method. Step 1. For our first sample we capture a predetermined number n 1 of catfish, say n 1 = 200. The fish are tagged and released unharmed back in the pond. Example 13.6Small Fish in a Big Pond

9. Excursions in Modern Mathematics, 7e: 13.5 - 9Copyright © 2010 Pearson Education, Inc. Step 2. After giving enough time for the released fish to mingle and disperse throughout the pond, we capture a second sample of n 2 catfish. While n 2 does not have to equal n 1, it is a good idea for the two samples to be of approximately the same order of magnitude. Let’s say that n 2 = 150. Of the 150 catfish in the second sample, 21 have tags (were part of the original sample). Example 13.6Small Fish in a Big Pond

10. Excursions in Modern Mathematics, 7e: 13.5 - 10Copyright © 2010 Pearson Education, Inc. Assuming the second sample is representative of the catfish population in the pond, the ratio of tagged fish in the second sample (21/150) is approximately the same as the ratio of tagged fish in the pond (200/N). This gives the approximate proportion 21/150 ≈ 200/N which in turn gives N ≈ 200 150/21 ≈ 1428.57 Example 13.6Small Fish in a Big Pond

11. Excursions in Modern Mathematics, 7e: 13.5 - 11Copyright © 2010 Pearson Education, Inc. Obviously, the value N = 1428.57 cannot be taken literally, since N must be a whole number. Besides, even in the best of cases, the computation is only an estimate. A sensible conclusion is that there are approximately N = 1400 catfish in the pond. Example 13.6Small Fish in a Big Pond