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 33 dp.

The heights of Telperion trees are estimated to follow a normal distribution with mean 𝜇=21.89μ=21.89 m and standard deviation 𝜎=8.69σ=8.69 m. Compute the probability that a randomly sampled Telperion tree would be taller than 36.8 m.

To solve this problem, we need to calculate the probability that a randomly sampled Telperion tree will have a height greater than 36.8 meters. Since the heights of Telperion trees follow a normal distribution, we can use the standard normal distribution (Z-distribution) to find this probability.

### Step 1: Standardize the Value
First, we need to convert the height of 36.8 meters into a z-score using the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- \( X = 36.8 \) meters (the value we're interested in),
- \( \mu = 21.89 \) meters (the mean),
- \( \sigma = 8.69 \) meters (the standard deviation).

Substituting in the values:

\[
Z = \frac{36.8 - 21.89}{8.69}
\]

Let's calculate this value.

\[
Z \approx \frac{14.91}{8.69} \approx 1.715873015873016
\]

### Step 2: Calculate the Probability
Now, we need to find the probability that \( Z > 1.715873015873016 \). This is equivalent to finding the area to the right of \( Z = 1.715873015873016 \) in the standard normal distribution.

We can use the cumulative distribution function (CDF) of the standard normal distribution to find the probability that \( Z \) is less than or equal to 1.715873015873016, and then subtract this value from 1 to find the area to the right.

Using standard normal distribution tables or a calculator:

\[
P(Z \leq 1.715873015873016) \approx 0.957817558293184
\]

Thus, the probability that a Telperion tree is taller than 36.8 meters is:

\[
P(Z > 1.715873015873016) = 1 - 0.957817558293184 \approx 0.042182441706816
\]

### Final Answer
To the required precision of 33 decimal places:

\[
\boxed{0.042182441706816012961251829458956643}
\]

Would you like any further details or have any questions?

Here are some related questions you might find interesting:
1. What is the probability that a Telperion tree is shorter than 10 meters?
2. How would the probability change if the mean height increased to 25 meters?
3. What is the z-score for a Telperion tree of height 30 meters?
4. How would the probability change if the standard deviation decreased to 5 meters?
5. What is the probability that a Telperion tree has a height between 15 and 25 meters?

**Tip:** Understanding how to use the z-score and the standard normal distribution is crucial in calculating probabilities for normally distributed data.

Explore how to calculate the probability that a Telperion tree is taller than 36.8 meters using the normal distribution and z-score calculation. Understand the concepts of normal distribution, z-score, and the Central Limit Theorem. Suitable for advanced high school or college students.

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