https://file.aiquickdraw.com/mathbot/file/1729478450377kas9v2mh.jpg

To solve the problem:

We are given that IQ scores are normally distributed with:
- Mean (μ) = 100
- Standard deviation (σ) = 20

The problem asks for the probability that a randomly selected adult has an IQ between 87 and 113.

### Step 1: Convert the IQ scores to z-scores
The formula to convert a value to a z-score is:

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

For \(x = 87\):
\[
z_1 = \frac{87 - 100}{20} = \frac{-13}{20} = -0.65
\]

For \(x = 113\):
\[
z_2 = \frac{113 - 100}{20} = \frac{13}{20} = 0.65
\]

### Step 2: Use the standard normal distribution table
Now we need to find the probability corresponding to these z-scores:
- The cumulative probability for \(z_1 = -0.65\) is approximately 0.2578.
- The cumulative probability for \(z_2 = 0.65\) is approximately 0.7422.

### Step 3: Calculate the probability between 87 and 113
The probability that an IQ is between 87 and 113 is:

\[
P(87 \leq x \leq 113) = P(z_2) - P(z_1) = 0.7422 - 0.2578 = 0.4844
\]

Thus, the probability that a randomly selected adult has an IQ between 87 and 113 is **0.4844**.

Would you like further details or clarification on any part of this? 

Here are five related questions to consider:
1. How would the probability change if the IQ range was between 85 and 115?
2. How do you find the probability for values not exactly on the z-score table?
3. What does a z-score represent in a normal distribution?
4. How do changes in the standard deviation affect the probabilities?
5. How do you interpret probabilities in real-world applications of normal distribution?

**Tip:** You can always visualize probabilities on a normal curve by shading the region of interest between two z-scores.

Assume that adults have IQ scores that are normally distributed with a mean of μ = 100 and a standard deviation σ = 20. Find the probability that a randomly selected adult has an IQ between 87 and 113.

Learn how to find the probability that a randomly selected adult has an IQ between 87 and 113, given a normal distribution with a mean of 100 and a standard deviation of 20. This problem covers key probability concepts, Z-scores, and the use of a standard normal distribution table.

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