Mathematics
Grade 7
15 min
Estimate population size using proportions
Estimate population size using proportions
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Introduction & Learning Objectives
Learning Objectives
Define key terms related to population estimation, such as 'population', 'sample', 'marked sample', and 'recaptured sample'.
Explain the purpose and basic steps of the capture-recapture method for estimating population size.
Set up a correct proportion to model the relationship between samples and the total population.
Solve proportions using cross-multiplication to find an unknown population size.
Interpret the estimated population size in the context of a real-world scenario.
Identify basic assumptions made when using the capture-recapture method.
Ever wondered how scientists count all the fish in a huge lake or all the deer in a vast forest without catching every single one? 🐟🦌 It seems impossible, right?
In t...
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Key Concepts & Vocabulary
TermDefinitionExample
PopulationThe entire group of individuals or items that you are interested in studying.All the squirrels living in a particular park.
SampleA smaller, representative group taken from a larger population.15 squirrels caught from the park to study their health.
ProportionAn equation that states two ratios are equal. It's often written as $\frac{a}{b} = \frac{c}{d}$.$\frac{1}{2} = \frac{5}{10}$ is a proportion because both ratios simplify to the same value.
Capture-Recapture MethodA technique used to estimate the size of a population by capturing, marking, and releasing a sample, then later capturing a second sample.Catching 20 fish, tagging them, releasing them, then later catching 30 fish and seeing how many of them have tags.
Marked Sample (First Capture)The ini...
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Core Formulas
The Capture-Recapture Proportion Formula
$\frac{\text{Number marked in 1st sample}}{\text{Total population (N)}} = \frac{\text{Number marked in 2nd sample}}{\text{Total in 2nd sample}}$
This formula sets up a proportion where the ratio of marked individuals to the total population is assumed to be the same as the ratio of marked individuals to the total in a second sample. We use it to solve for 'N', the unknown total population size.
Cross-Multiplication Rule for Proportions
If $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$.
This rule is used to solve for an unknown variable in a proportion. You multiply the numerator of one ratio by the denominator of the other ratio, and set the products equal to each other.
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Challenging
What is the most likely effect on the population estimate if a significant number of marked animals lose their tags before the second capture?
A.The estimate will be higher than the actual population size (overestimate).
B.The estimate will be lower than the actual population size (underestimate).
C.The estimate will not be affected.
D.The estimate will be exactly half of the actual population size.
Challenging
Some animals become 'trap-happy,' meaning they are more likely to be caught again after being captured once. If this happens, what is the likely effect on the population estimate?
A.The estimate will be higher than the actual population size (overestimate).
B.The estimate will be lower than the actual population size (underestimate).
C.The estimate will be perfectly accurate.
D.The estimate will be zero.
Challenging
In which of the following scenarios would the capture-recapture method be most UNRELIABLE for estimating a bird population?
A.study of birds on a remote island with no predators.
B.study of birds in a small, enclosed sanctuary.
C.study of birds in a forest where the marks (leg bands) are permanent.
D.study of birds at a location along a major migratory route.
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