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The problem involves finding the area of the shaded region under a normal distribution curve.

Given:
- The IQ scores are normally distributed with:
  - **Mean (μ) = 100**
  - **Standard deviation (σ) = 15**
- The shaded region represents IQ scores below **88**.

We need to calculate the probability that a randomly selected score is less than 88 (i.e., the area under the curve to the left of 88).

### Steps:
1. **Standardize the value**: First, we compute the z-score for the value 88 using the formula:

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

   where \( x = 88 \), \( \mu = 100 \), and \( \sigma = 15 \).

   \[
   z = \frac{88 - 100}{15} = \frac{-12}{15} = -0.8
   \]

2. **Find the cumulative probability**: Using a standard normal distribution table or a calculator, find the cumulative probability corresponding to \( z = -0.8 \).

   The cumulative probability (area to the left of \( z = -0.8 \)) is approximately **0.2119**.

Thus, the area of the shaded region is **0.2119**.

Would you like a more detailed breakdown or further explanation? Let me know!

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Here are five related questions to explore further:
1. How would the area change if the standard deviation was larger?
2. What is the probability that a randomly selected IQ score is greater than 88?
3. How do you interpret z-scores in a real-world context?
4. What is the probability for a range of IQ scores, say between 88 and 100?
5. How would you calculate the area for values above a certain IQ?

**Tip**: Z-scores help standardize different normal distributions, making them comparable across various datasets.

Find the area of the shaded region. 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 shaded region represents scores below 88.

This problem explores how to calculate the area under a normal distribution curve for IQ scores. With a mean of 100 and a standard deviation of 15, the shaded region corresponds to scores below 88. Learn how to standardize the value and use the cumulative probability function to find the area of the shaded region.

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