Question

The ages of trees in a forest are normally distributed with a mean of 25 years and a standard deviation of 5 years usibg the empricl rule Approximately what percent of the trees are between 20 and 30 years old?

Answer 1

68.3% of the trees are between 20 and 30 years old.

Explanation:

We are given that the ages of trees in a forest are normally distributed with a mean of 25 years and a standard deviation of 5 years.

Let X = ages of trees in a forest

So, X ~ N(
`\mu = 25 , \sigma^(2)=5^(2)`)

The standard normal z score distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

So, percent of the trees between 20 and 30 years old is given by;

P(20 < X < 30) = P(X < 30) - P(X <= 20)

P(X < 30) = P(
`(X-\mu)/(\sigma)` <
`(30-25)/(5)` ) = P(Z < 1) = 0.84134 {using z table}

P(X <= 20) = P(
`(X-\mu)/(\sigma)` <=
`(20-25)/(5)` ) = P(Z <= -1) = 1 - P(Z < 1)= 1 - 0.84134 = 0.15866

So, P(20 < X < 30) = 0.84134- 0.15866 = 0.68268 or 68.3%

Therefore, 68.3% of the trees are between 20 and 30 years old.