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 area of the shaded region is 

enter your response here. 

​(Round to four decimal places as​ needed.) 

the shaded region is in the middle of the slope beween the number 85 and 110

To find the area of the shaded region between IQ scores 85 and 110, we can use the properties of the normal distribution.

Given:
- Mean \( \mu = 100 \)
- Standard deviation \( \sigma = 15 \)
- The IQ scores 85 and 110 define the boundaries of the shaded region.

The area under the normal curve between these two points represents the probability that an individual has an IQ score between 85 and 110. We calculate this probability by finding the **z-scores** for the boundaries and then finding the cumulative probability between these z-scores.

### Step 1: Calculate the z-scores

The z-score formula is:
\[
z = \frac{x - \mu}{\sigma}
\]

For \( x = 85 \):
\[
z_1 = \frac{85 - 100}{15} = \frac{-15}{15} = -1
\]

For \( x = 110 \):
\[
z_2 = \frac{110 - 100}{15} = \frac{10}{15} = 0.6667
\]

### Step 2: Find the cumulative probabilities

Using a standard normal distribution table or a calculator:

- The cumulative probability for \( z_1 = -1 \) is approximately 0.1587.
- The cumulative probability for \( z_2 = 0.6667 \) is approximately 0.7475.

### Step 3: Find the area of the shaded region

The area of the shaded region between these z-scores is the difference between the cumulative probabilities:

\[
\text{Area} = P(z_2) - P(z_1) = 0.7475 - 0.1587 = 0.5888
\]

### Final Answer:
The area of the shaded region is **0.5888** (rounded to four decimal places).

Would you like any further explanation or additional details on this? 

Here are 5 related questions to expand your understanding:
1. How would the area change if the boundaries were 90 and 115?
2. What is the significance of the z-score in a normal distribution?
3. Can you calculate the area for an IQ score less than 85?
4. How would the area be affected if the standard deviation were 20 instead of 15?
5. What does the area under the normal curve represent in real-world terms?

**Tip:** Understanding the z-score helps in quickly determining how far a particular value is from the mean in standard deviations.

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 area of the shaded region is (Round to four decimal places as needed.) The shaded region is in the middle of the slope between the numbers 85 and 110.

Learn how to find the area of the shaded region between IQ scores 85 and 110 using normal distribution properties. This problem involves calculating z-scores and using cumulative probabilities to determine the area under the normal curve. Suitable for students in Grades 11-12.

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