A distribution of values is normal with a mean of 65 and a standard deviation of 59.1. Use exact z-scores or z-scores rounded to 2 decimal places.

Find the probability that a randomly selected value is greater than 200.9.
P(X > 200.9) = 


Enter your answer as a number accurate to 4 decimal places.

To find the probability that a randomly selected value is greater than 200.9 in a normal distribution with a mean (\(\mu\)) of 65 and a standard deviation (\(\sigma\)) of 59.1, we will use the **z-score formula**:

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

Where:
- \( X = 200.9 \)
- \( \mu = 65 \)
- \( \sigma = 59.1 \)

### Step 1: Calculate the z-score
\[
z = \frac{200.9 - 65}{59.1} = \frac{135.9}{59.1} \approx 2.30
\]

### Step 2: Use the z-score to find the probability
Now, we need to find \( P(X > 200.9) \), which corresponds to \( P(Z > 2.30) \).

From a z-score table or using a calculator, the probability for \( Z \leq 2.30 \) is approximately 0.9893. Since we want \( P(Z > 2.30) \), we subtract this value from 1:

\[
P(X > 200.9) = 1 - 0.9893 = 0.0107
\]

### Final Answer:
\[
P(X > 200.9) \approx 0.0107
\]

Would you like further explanation, or have any questions?

Here are 5 related questions to explore this topic further:
1. How do we interpret z-scores in the context of a normal distribution?
2. How can we find the probability for values less than a certain point?
3. What is the empirical rule, and how does it apply to normal distributions?
4. How do we calculate probabilities for values between two points in a normal distribution?
5. How does the standard deviation affect the shape of a normal distribution?

**Tip**: The z-score indicates how many standard deviations a data point is from the mean, which helps in determining its relative position in a normal distribution.

A distribution of values is normal with a mean of 65 and a standard deviation of 59.1. Use exact z-scores or z-scores rounded to 2 decimal places. Find the probability that a randomly selected value is greater than 200.9. P(X > 200.9) = 

Learn how to calculate the probability that a randomly selected value is greater than 200.9 in a normal distribution with a mean of 65 and a standard deviation of 59.1. This solution involves using z-scores and standard normal distribution tables, making it suitable for students in Grades 10-12.

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