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Part 1

Assume the random variable x is normally distributed with mean 

muμequals=8888 

and standard deviation 

sigmaσequals=44. 

Find the indicated probability.

To find the indicated probability for a normally distributed random variable \( x \) with a given mean \( \mu \) and standard deviation \( \sigma \), you can follow these steps:

1. **Standardize the random variable**: Convert the given \( x \) value to a standard normal variable \( z \) using the formula:
   \[
   z = \frac{x - \mu}{\sigma}
   \]

2. **Use the z-table**: Find the corresponding probability from the z-table for the computed \( z \)-score.

Let's assume we need to find the probability \( P(X \leq x) \) for a specific value of \( x \). For demonstration purposes, let’s find the probability for \( x = 90 \).

Given:
- Mean (\( \mu \)) = 88
- Standard deviation (\( \sigma \)) = 4

### Step-by-Step Solution:

1. **Standardize the \( x \) value**:
   \[
   z = \frac{x - \mu}{\sigma} = \frac{90 - 88}{4} = \frac{2}{4} = 0.5
   \]

2. **Find the corresponding probability**:
   - Look up the \( z \)-score of 0.5 in the z-table.
   - The cumulative probability for \( z = 0.5 \) is approximately 0.6915.

Therefore, the probability \( P(X \leq 90) \) is approximately 0.6915.

If you have a specific \( x \) value in mind, please provide that value, and I can find the corresponding probability for you.

Do you want more details or have any questions? Here are some related questions for further exploration:

1. What is the probability \( P(X \leq 85) \) for the given normal distribution?
2. How do you find the probability \( P(X > 92) \)?
3. What is the probability \( P(86 \leq X \leq 90) \)?
4. How do you use the standard normal distribution to find probabilities for non-standard normal distributions?
5. What is the probability that \( X \) is within one standard deviation of the mean?
6. Can you explain the empirical rule (68-95-99.7 rule) for normal distributions?
7. How do you find the \( x \) value corresponding to a given percentile in a normal distribution?
8. What is the probability \( P(X < 80) \) for the given normal distribution?

**Tip**: When working with probabilities for normal distributions, always convert your raw scores to \( z \)-scores to use the standard normal table effectively.

Learn how to calculate probabilities in a normal distribution with mean μ = 88 and standard deviation σ = 4. This problem involves standardizing the random variable x and using the z-table to find probabilities. Suitable for advanced high school students and anyone interested in statistics.

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