IDNLearn.com is your go-to platform for finding accurate and reliable answers. Whether your question is simple or complex, our community is here to provide detailed and trustworthy answers quickly and effectively.
Sagot :
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.
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.
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover the answers you need at IDNLearn.com. Thanks for visiting, and come back soon for more valuable insights.