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To find the area of the shaded region representing IQ scores between 85 and 125, given that the scores are normally distributed with a mean (\(\mu\)) of 100 and a standard deviation (\(\sigma\)) of 15, we can calculate the probability of scores falling between 85 and 125.

Here’s the process:

1. **Standardize the bounds**:
   - Convert the values 85 and 125 to \(z\)-scores using the formula:
     \[
     z = \frac{X - \mu}{\sigma}
     \]
   - For \(X = 85\):
     \[
     z_{85} = \frac{85 - 100}{15} = -1
     \]
   - For \(X = 125\):
     \[
     z_{125} = \frac{125 - 100}{15} = 1.67
     \]

2. **Find the corresponding probabilities**:
   - Using standard normal distribution tables or a calculator:
     - \(P(Z \leq -1) \approx 0.1587\)
     - \(P(Z \leq 1.67) \approx 0.9525\)

3. **Calculate the area**:
   - The area of the shaded region is the difference between these two probabilities:
     \[
     P(85 \leq X \leq 125) = P(Z \leq 1.67) - P(Z \leq -1)
     \]
     \[
     \approx 0.9525 - 0.1587 = 0.7938
     \]

So, the area of the shaded region is approximately **0.7938**.

Would you like a more detailed explanation of any step?

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### Related Questions:
1. How do you interpret a \(z\)-score in the context of a normal distribution?
2. What is the probability of an IQ score being below 85 or above 125 in this distribution?
3. How would the area change if the standard deviation were different?
4. Can we use a calculator to find the area directly?
5. Why do we standardize using \(z\)-scores in normal distribution problems?

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**Tip:** When working with normal distributions, sketching the \(z\)-score positions on a standard bell curve can help visualize the problem and improve accuracy.

Find the area of the shaded region. The graph depicts IQ scores of adults, normally distributed with a mean of 100 and a standard deviation of 15. Calculate the area between scores 85 and 125.

Learn to find the area under the normal distribution curve between IQ scores 85 and 125, given a mean of 100 and a standard deviation of 15. This involves calculating Z-scores and using standard normal tables. Suitable for high school students in Grades 11-12.

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