Find the indicated IQ score. the graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. The x-value is 0.0808

To find the indicated IQ score corresponding to the given \(x\)-value (probability), we need to use the properties of the normal distribution.

Given:
- Mean (\(\mu\)) = 100
- Standard deviation (\(\sigma\)) = 15
- \(x\)-value (probability) = 0.0808

### Steps to Solve:

1. **Understand the Problem:**
   The \(x\)-value provided is the cumulative probability up to a certain IQ score. We want to find the IQ score that corresponds to this cumulative probability in a normal distribution.

2. **Find the Z-score Corresponding to the Probability:**
   A standard normal distribution (Z-distribution) has a mean of 0 and a standard deviation of 1. The cumulative probability of 0.0808 corresponds to a certain Z-score, which we can find using the inverse of the standard normal distribution (also called the Z-table or the inverse normal function).

3. **Convert the Z-score to the IQ Score:**
   The formula to convert a Z-score to a value in a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) is:

   \[
   X = \mu + Z \cdot \sigma
   \]

### Calculation:

1. Find the Z-score for which the cumulative probability is 0.0808.
   
2. Use that Z-score to calculate the IQ score.

Let's calculate the Z-score and the corresponding IQ score.The IQ score corresponding to the given cumulative probability of 0.0808 is approximately **79.00**.

### Summary:
- The Z-score associated with a cumulative probability of 0.0808 is used to find the IQ score in a normal distribution with a mean of 100 and a standard deviation of 15.
- The calculated IQ score is approximately 79.00.

Would you like more details on any step, or do you have any other questions?

### Related Questions:
1. How do you find a Z-score given a percentile?
2. What is the inverse normal distribution function, and how is it used?
3. How can you calculate probabilities for a normal distribution?
4. What are some real-world applications of normal distribution?
5. How do mean and standard deviation affect the shape of a normal distribution graph?

### Tip:
For problems involving normal distributions, understanding the properties of Z-scores and how to use cumulative probability tables or functions is crucial for finding exact values.

Find the indicated IQ score. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. The x-value is 0.0808.

Learn how to find the IQ score corresponding to a cumulative probability of 0.0808 in a normal distribution with a mean of 100 and a standard deviation of 15. This problem involves Z-score calculations and inverse normal distribution. Suitable for students in Grades 10-12.

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