Estimate population size using proportions

Full solution

Q. Recently, frogs have been disappearing from the wild due to a contagious fungus. To monitor the current population in a certain region, Alan collects

28

frogs, marks them, and releases them. Later, he collects

110

frogs and finds that

11

of them are marked. To the nearest whole number, what is the best estimate for the frog population?

\_\_\_\_

Given Information: We are given the following information: $\newline$ - Alan marks and releases $28$ frogs. $\newline$ - Later, he collects $110$ frogs, out of which $11$ are marked. $\newline$ We need to estimate the total frog population in the region based on this data. $\newline$ We will use the capture-recapture method, which assumes that the proportion of marked frogs in the second sample represents the same proportion of the total population.
Proportion Setup: Set up a proportion to represent the relationship between the marked and unmarked frogs in the second sample and the total population. $\newline$ Let $p$ be the estimated frog population. $\newline$ The proportion is as follows: $\newline$ $\frac{\text{marked frogs in second sample}}{\text{total frogs in second sample}} = \frac{\text{total marked frogs}}{\text{estimated total population}}$ $\newline$ $\frac{11}{110} = \frac{28}{p}$
Cross Multiplication: Cross multiply to solve for $p$ : $11 \times p = 28 \times 110$
Solving for p: Now, divide both sides of the equation by $11$ to solve for $p$ : $\newline$ $(11 \cdot p) / 11 = (28 \cdot 110) / 11$ $\newline$ $p = (28 \cdot 110) / 11$
Calculating p: Calculate the value of p: $\newline$ p = $(28 \times 110) / 11$ $\newline$ p = $3080 / 11$ $\newline$ p = $280$