Question

A forestry study found that the diameter of the trees in a forest is normally distributed with mean 34 cm with a standard deviation of 8 cm. A group of 4 trees will be used as timber if the average of the 4 trees diameter is not too thick or thin. Specifically it is desired for the mean diameter to be between 30 and 40 cm in diameter. Find the probability that a randomly chosen group of 4 trees can be used as timber

Answer 1

The probability that a randomly selected group of four trees can be used as timber is 4.5 × 10⁻⁵

Explanation:

The given parameters are;

Mean = 34 cm

The standard deviation = 8 cm

The mean

The Z score is
`Z=(x-\mu )/(\sigma )`, which gives;

For x = 30 we have;

`Z=(30-34 )/(8 ) = -0.5`

P(x>30) = 1 - 0.30854 = 0.69146

For x = 40, we have

`Z=(40-34 )/(8 ) = 0.75`

P(x < 40) = 0.77337

Therefore, the probability that the mean of four trees is between 30 and 40 is given as follows;

P(30 < x < 40) = 0.77337 - 0.69146 = 0.08191

The probability that a randomly selected group of four trees can be used as timber is given as follows;

Binomial distribution

`P(X = 4) = \dbinom{4}{4} \left (0.08191\right )^(4)\left (1-0.08191 \right )^(0) = 4.5 * 10^(-5)`