To determine the percentage of pine trees taller than 100 feet, we need to follow a systematic approach that includes finding the z-score and using the z-score table to find the cumulative probability to the left of the z-score. Here are the steps:
1. Find the z-score:
The z-score is a measure of how many standard deviations away a particular value is from the mean. It is calculated using the formula:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where:
- [tex]\(X\)[/tex] is the value we are interested in (in this case, 100 feet),
- [tex]\(\mu\)[/tex] is the mean (86 feet),
- [tex]\(\sigma\)[/tex] is the standard deviation (8 feet).
Plugging in the values:
[tex]\[
z = \frac{100 - 86}{8} = \frac{14}{8} = 1.75
\][/tex]
2. Find the cumulative probability:
Using the z-score table provided, we locate the value corresponding to a z-score of 1.75. According to the table, for a z-score of 1.75, the cumulative probability to the left is approximately 0.9599. This means that 95.99% of the pine trees are 100 feet tall or shorter.
3. Calculate the percentage above the specified height:
To find the percentage of pine trees taller than 100 feet, we subtract the cumulative probability from 1 (since the total probability is 1 or 100%).
[tex]\[
\text{Percentage above} = (1 - \text{cumulative probability}) \times 100
\][/tex]
[tex]\[
\text{Percentage above} = (1 - 0.9599) \times 100 = 0.0401 \times 100 = 4.01\%
\][/tex]
Since we need to select from the given choices, and the closest option is 4%, we can approximate to:
Answer: A. 4%
Therefore, approximately 4% of the pine trees in the forest are taller than 100 feet.
