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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. Round answer to four decimal places as needed.

The image shows a normal distribution representing IQ scores, with a mean of 100 and a standard deviation of 15. The shaded region appears to correspond to IQ scores greater than 110.

### Step-by-Step Solution:

1. **Convert IQ score to a z-score:**
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   Where:
   - \(X = 110\) (the IQ score)
   - \(\mu = 100\) (the mean IQ score)
   - \(\sigma = 15\) (the standard deviation)

   Plugging the values in:
   \[
   z = \frac{110 - 100}{15} = \frac{10}{15} \approx 0.6667
   \]

2. **Find the area to the right of \(z = 0.6667\):**
   Using the positive z-score table, we need to find the area to the right of this z-score.

   From the z-table, the cumulative area to the left of \(z = 0.6667\) is approximately **0.7475**.

3. **Calculate the area of the shaded region:**
   The shaded area is the area to the right of \(z = 0.6667\), which is:
   \[
   1 - 0.7475 = 0.2525
   \]

### Final Answer:
The area of the shaded region is approximately **0.2525**.

Would you like further details or have any questions?

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Here are five related questions:
1. How is a z-score calculated for different IQ values?
2. What does the area under the normal curve represent in terms of probability?
3. How would the area change if the IQ score were 120 instead of 110?
4. What is the significance of the standard deviation in a normal distribution?
5. How can we interpret a z-score of 0.6667 in terms of standard deviations from the mean?

**Tip:** When using a standard normal distribution table, make sure to check whether you're looking for the area to the left or right of the z-score.

Solve a problem involving a normal distribution of IQ scores with a mean of 100 and standard deviation of 15. Learn how to find the area of the shaded region greater than an IQ of 110 using z-scores and a normal distribution table. Suitable for high school students studying statistics.

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