Question

An ornithologist who specializes in cockatoos in Australia decides to estimate the cockatoo population in a certain region. To do so, he traps 54 cockatoos and marks them. After releasing the cockatoos and waiting a bit, he traps 290 cockatoos and observes that 29 of them are marked. To the nearest whole number, what is the best estimate for the cockatoo population?

Answer 1

the population size is 540 cockatoos

Explanation:

Denoting P as the total population , since the person took the 54 cockatoos and mark them , after he released them there are a proportion of marked cockatoos in the population equal to

proportion of marked cockatoos = marked cockatoos / total number of cockatoos = 54 / P

then if he takes a sample , if we assume that the marked cockatoos are well mixed around the population and each cockatoo has the same probability of being trapped , then if we take cockatoos at random , is almost likely that he traps cockatoos in the same proportion , then

proportion of marked cockatoos = 29/290 = 54/P

P= 54 * 290 / 29 = 540 cockatoos

Note

- Mathematically , is the same that saying that each sample has the the same probability of being chosen.

- Actually if the traps 290 cockatoos out of 540 , the actual probability can be calculated through an hypergeometric distribution whose most probable value of the population size is 540 cockatoos

Answer 2

540

Explanation:

Let the estimate total population=y

Initially, out of a total of y, 54 are marked.

Then out of a sample of 290 cockatoos, 29 of them are marked.

We take the ratio of the population to the sample and do same for the number of marked in each category.

y:290 = 54: 29

[TeX]\frac{y}{290}=\frac{54}{29}[/TeX]

Cross multiplying

y X 29 = 290 X 54

Divide both sides by 29 to obtain y.

[TeX]\frac{y X 29}{29}=\frac{290 X 54}{29}[/TeX]

y= 54 X 10 =540

The best estimate of the cuckatoo population is 540