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To solve this problem, we’ll use the concept of the standard normal distribution.

### Problem Summary
- **Average age (mean, μ)** = 59 years
- **Standard deviation (σ)** = 3 years
- **Range**: Find the probability that a randomly selected CEO is between 56 and 62 years old.

### Step 1: Convert Ages to Z-scores
The Z-score formula is:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \( X \) is the age, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

1. **For age 56**:
   \[
   Z = \frac{56 - 59}{3} = \frac{-3}{3} = -1
   \]

2. **For age 62**:
   \[
   Z = \frac{62 - 59}{3} = \frac{3}{3} = 1
   \]

So, we need to find the probability that \( Z \) is between -1 and 1.

### Step 2: Use Standard Normal Distribution Table
The probability that \( Z \) is between -1 and 1 is the area under the standard normal curve from \( Z = -1 \) to \( Z = 1 \).

- From a standard normal distribution table:
  - **P(Z ≤ 1)** ≈ 0.8413
  - **P(Z ≤ -1)** ≈ 0.1587

To find the probability between these Z-scores:
\[
P(-1 \leq Z \leq 1) = P(Z \leq 1) - P(Z \leq -1)
\]
\[
= 0.8413 - 0.1587 = 0.6826
\]

### Step 3: Convert Probability to Percentage
\[
0.6826 \times 100 \approx 68.26\%
\]

### Answer
The probability that a randomly selected CEO will be between 56 and 62 years old is approximately **68.26%**.

Would you like more details, or do you have any questions?

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Here are some related questions:
1. What is the probability of a randomly selected CEO being younger than 56 years?
2. How do you calculate the Z-score if the range changes to 55 and 63 years?
3. What is the probability of a CEO being older than 62 years?
4. How does increasing the standard deviation affect the probability?
5. Can we use the same approach for non-normal distributions?

**Tip**: Z-scores allow easy comparison of values from different distributions by normalizing to a standard scale.

The average age of CEOs is 59 years. Assume the random variable is normally distributed. If the standard deviation is 3 years, find the probability that the age of a randomly selected CEO will be between 56 and 62 years old.

Calculate the probability of a randomly selected CEO's age falling between 56 and 62 years, given a mean age of 59 and a standard deviation of 3 years. This problem uses Z-scores and the standard normal distribution, perfect for high school statistics students.

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