To determine the probability that a randomly selected tree in the forest has a height greater than or equal to 37 meters, we can follow these steps:
### Step-by-Step Solution:
1. Identify the given values:
- Mean height ([tex]\(\mu\)[/tex]) = 25 meters
- Standard deviation ([tex]\(\sigma\)[/tex]) = 6 meters
- Tree height ([tex]\(X\)[/tex]) = 37 meters
2. Calculate the z-score for the tree height of 37 meters:
- The z-score formula is given by: [tex]\( z = \frac{X - \mu}{\sigma} \)[/tex]
- Substituting the given values:
[tex]\[
z = \frac{37 - 25}{6} = \frac{12}{6} = 2.0
\][/tex]
3. Find the cumulative probability for the z-score from the standard normal table:
- We look up the probability for [tex]\( z = 2.0 \)[/tex] in the provided table:
[tex]\[
\text{Probability for } z = 2.0 \text{ is } 0.9772
\][/tex]
4. Interpret the cumulative probability:
- The cumulative probability of 0.9772 represents the probability that a tree is shorter than 37 meters.
5. Compute the probability that a tree is taller than or equal to 37 meters:
- Since the cumulative probability represents [tex]\( P(X < 37) \)[/tex], we need [tex]\( P(X \geq 37) \)[/tex]:
[tex]\[
P(X \geq 37) = 1 - P(X < 37) = 1 - 0.9772 = 0.0228
\][/tex]
6. Convert this probability to a percentage:
- We multiply the decimal probability by 100 to get the percentage:
[tex]\[
0.0228 \times 100 = 2.28\%
\][/tex]
Therefore, the probability that a randomly selected tree in the forest has a height greater than or equal to 37 meters is 2.28%.
Among the provided options, the one closest to 2.28% is:
[tex]\[
\boxed{3\%}
\][/tex]
