X is a normally distributed random variable with mean 56 and standard deviation 17.
What is the probability that X is between 25 and 87?

To solve this problem, we need to find the probability that a normally distributed random variable \( X \) with a mean \( \mu = 56 \) and a standard deviation \( \sigma = 17 \) falls between 25 and 87.

### Step 1: Convert to the Standard Normal Distribution
We need to convert the values 25 and 87 into their corresponding z-scores using the formula:

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

- For \( x = 25 \):
\[
z_1 = \frac{25 - 56}{17} = \frac{-31}{17} \approx -1.82
\]

- For \( x = 87 \):
\[
z_2 = \frac{87 - 56}{17} = \frac{31}{17} \approx 1.82
\]

### Step 2: Find the Probability
The probability that \( X \) is between 25 and 87 is equivalent to finding the area under the standard normal curve between \( z_1 \) and \( z_2 \).

Using standard normal distribution tables or a calculator:

- The cumulative probability corresponding to \( z_1 = -1.82 \) is approximately 0.0344.
- The cumulative probability corresponding to \( z_2 = 1.82 \) is approximately 0.9656.

### Step 3: Calculate the Desired Probability
The probability that \( X \) is between 25 and 87 is:

\[
P(25 \leq X \leq 87) = P(z_2) - P(z_1) = 0.9656 - 0.0344 = 0.9312
\]

### Final Answer:
The probability that \( X \) is between 25 and 87 is approximately **0.9312**, or **93.12%**.

Would you like more details or have any questions?

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Here are five related questions for further exploration:
1. What is the probability that \( X \) is greater than 87?
2. What is the probability that \( X \) is less than 25?
3. How do you calculate the z-score for any given value of \( X \)?
4. What is the probability that \( X \) is between 40 and 70?
5. How does the standard deviation affect the width of the normal distribution curve?

**Tip:** When dealing with normally distributed variables, always convert to z-scores for easier probability calculations using the standard normal table.

Discover how to calculate the probability of a normally distributed random variable X with mean 56 and standard deviation 17 being between 25 and 87. This problem involves the normal distribution, z-scores, and probability calculations. Suitable for advanced high school students and college-level learners.

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