A forestry service worker randomly selected 80 oak trees in a state park containing 2,000 … — SAT

A forestry service worker randomly selected $80$ oak trees in a state park containing $2,000$ oak trees. The worker found that $12$ of the selected oak trees showed signs of a leaf disease. Based on this sample, what is the estimated number of oak trees in the entire state park that show signs of the leaf disease?

Answer: 300

Answer

300

Because the sample of

80

oak trees was selected at random, it can be assumed to be representative of the population of

2,000

oak trees in the state park. The proportion of diseased trees in the sample is

\frac{12}{80} = 0.15

. Multiplying this proportion by the total population of

2,000

oak trees yields the estimated total number of diseased trees:

0.15 \times 2,000 = 300

Step-by-Step Solution

Calculate the sample proportion of diseased trees.

0.15

Dividing the

12

diseased trees by the sample size of

80

trees gives the proportion of diseased trees in the sample:

\frac{12}{80} = 0.15

Generalize the sample proportion to the entire population of oak trees in the park.

300

Multiplying the sample proportion of

0.15

by the total population of

2,000

oak trees yields the estimated total number of diseased trees:

0.15 \times 2,000 = 300

Key Concept

Generalizing results from a representative random sample to estimate a population parameter.

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