Turtle Mark-Recapture: How Much Confidence Does 300 Support?
Students estimate the size of a turtle population in a 2-hectare pond. On day 1 they capture 40 turtles, mark them, and release them. On day 8 they capture 60 turtles and find 8 marked. The class calculates N = (40 × 60) / 8 = 300 turtles and concludes the pond is healthy because the estimate exceeds a management threshold of 200. A local naturalist notes that turtles were held in shaded tubs for up to 90 minutes before marking, that some marks rubbed off within a week, and that heavy rain between sampling days raised the pond level and connected it to a ditch. The students' report also lists no measure of variability and does not say whether sampling was random. The class wants to know whether it can defend the conclusion at a school board meeting.
1. For each of the following, state whether it would tend to make the estimate too high, too low, or uncertain, and explain why: mark loss, trap-happy behavior, and immigration from the ditch.
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Mark loss makes the estimate too high. If marks rub off, some marked turtles are counted as unmarked on recapture, so the proportion marked in the second sample, 8/60, is lower than the true proportion. Since the estimate is the product of the two sample sizes divided by the number of marked recaptures, a smaller denominator inflates the result. Trap-happy behavior makes the estimate too low if marked turtles are more likely to be caught again, because the proportion marked on recapture rises and the denominator increases. Immigration has an uncertain net effect but tends to inflate the estimate: new unmarked turtles from the ditch lower the marked proportion and increase the estimated population, and the population is no longer closed. Holding turtles for 90 minutes could cause mortality or alter behavior, adding further uncertainty. The net direction depends on which violation dominates, which is why the report should not present 300 as an exact count.
2. Propose three specific improvements to the sampling protocol that would make the estimate more defensible, and explain what assumption each improvement addresses.
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Use a permanent or double-marking scheme, such as a notch plus a visible paint mark, so mark loss can be detected and estimated; this addresses the assumption that marks are not lost. Reduce holding time and release turtles near their capture point to limit mortality and behavioral change; this addresses the assumption that marked and unmarked animals mix and survive equally. Sample randomly across the pond and at multiple recapture occasions rather than one recapture, using a Schnabel or closed-capture model; this addresses the assumption that every animal has an equal chance of capture and that the population is closed. If the pond connects to the ditch, sample when it is isolated or treat the ditch as part of the sampled area. Report a confidence interval and the raw counts. These changes do not guarantee compliance, but they make the assumptions testable rather than hidden.
3. Even if the point estimate were accepted, would the data support the claim that the population exceeds the management threshold of 200? Use the recapture count to justify your answer.
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Not confidently. The estimate has substantial sampling error because only 8 marked turtles were recaptured. Treating the recapture count as approximately Poisson with mean 8, an exact 95% interval is roughly 3.5 to 15.8 marked recaptures. Converting with N = (40 × 60)/R gives approximate bounds of about 690 turtles at R = 3.5 and about 150 turtles at R = 15.8. That interval includes values below the 200 threshold, so the data do not establish that the population exceeds 200. The point estimate of 300 is compatible with a population of about 200 or fewer. A defensible statement is that the pond likely contains a few hundred turtles, but the management threshold cannot be evaluated without better marking, a closed-population design, and repeated samples. Claiming the pond is healthy is an overreach.