Question

Proportion of Tall Trees:

The average height of all trees in a normally distributed lodgepole pine plantation is 15 meters, with a variance of 9 square meters. Determine the proportion of trees in this plantation that are greater than 16 meters in height.

Answer 1

To determine the proportion of trees greater than 16 meters in height in a normally distributed plantation, use the z-score formula and the standard normal distribution. With an average height of 15 meters and a variance of 9 square meters, the proportion of trees greater than 16 meters is approximately 37.07%.

Step-by-step explanation:

To determine the proportion of trees in the plantation that are greater than 16 meters in height, we need to use the normal distribution. The average height of the trees is 15 meters, with a variance of 9 square meters.

First, we need to find the standard deviation by taking the square root of the variance. √9 = 3 meters.

Next, we calculate the z-score of a tree with a height of 16 meters using the formula: z = (x - µ) / σ, where x is the height of the tree, µ is the mean, and σ is the standard deviation.

So, z = (16 - 15) / 3 = 1/3.

Using a standard normal distribution table or calculator, we find that the proportion of trees with a height greater than 16 meters is approximately 0.3707, or 37.07%.