Answer the following question, keeping in mind that your answer must be a number between 0 and 1, i.e., do not use percentages. Give your final answer with at least 3
dp.

The heights of Telperion trees are estimated to follow a normal distribution with mean μ=45.54
m and standard deviation σ=8.43
m. Compute the probability that a randomly sampled Telperion tree would be taller than 50.03 m.

To find the probability that a randomly sampled Telperion tree is taller than 50.03 meters, we need to use the properties of the normal distribution. The problem provides that the heights of Telperion trees follow a normal distribution with mean \( \mu = 45.54 \) m and standard deviation \( \sigma = 8.43 \) m.

### Step 1: Compute the z-score
The z-score is calculated using the formula:
\[
z = \frac{x - \mu}{\sigma}
\]
where:
- \( x = 50.03 \) m is the height we're interested in,
- \( \mu = 45.54 \) m is the mean,
- \( \sigma = 8.43 \) m is the standard deviation.

Substituting in the values:
\[
z = \frac{50.03 - 45.54}{8.43} = \frac{4.49}{8.43} \approx 0.5324
\]

### Step 2: Find the probability
The z-score tells us how many standard deviations above the mean the height of 50.03 m is. We now need to find the probability that a Telperion tree is taller than 50.03 m, which is equivalent to finding the area to the right of \( z = 0.5324 \) under the standard normal distribution.

Using standard normal distribution tables or a calculator, the probability corresponding to \( z = 0.5324 \) is approximately 0.7031. This value represents the area to the left of the z-score.

The probability that a tree is taller than 50.03 m is the area to the right of this z-score:
\[
P(X > 50.03) = 1 - 0.7031 = 0.2969
\]

### Final Answer
The probability that a randomly sampled Telperion tree would be taller than 50.03 m is **0.297** (to three decimal places).

---

Would you like more details or have any questions? Here are some related questions:

1. What is the probability that a tree is shorter than 40 m?
2. How do you calculate the probability that a tree height falls within a certain range?
3. How does changing the mean height affect the probability?
4. What is the impact of increasing the standard deviation on this probability?
5. How can you use a normal distribution to estimate the percentage of trees within a certain height?

**Tip:** When dealing with normal distributions, converting to a z-score simplifies finding probabilities, as it standardizes different normal distributions to a single standard normal distribution.

Learn how to calculate the probability that a randomly sampled Telperion tree is taller than 50.03 meters using normal distribution properties. This problem involves computing z-scores, understanding standard deviations, and applying probability concepts.

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