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Let's go through each part of the question one by one.

### Part a: Distribution of X
The problem says that the average student loan debt is $25,250, and the standard deviation is $10,600. The random variable $ X $ represents the student loan debt of a randomly selected college graduate, and it follows a normal distribution, which is written as:
\[
X \sim N(\mu, \sigma^2)
\]
Where $ \mu $ is the mean and $ \sigma^2 $ is the variance (the square of the standard deviation).

- Mean $ \mu = 25,250 $
- Standard deviation $ \sigma = 10,600 $

Thus, the distribution is:
\[
X \sim N(25250, 10600^2)
\]

### Part b: Probability that loan debt is between $9,100 and $22,850
We need to find the probability $ P(9100 < X < 22850) $. To do this, we standardize the values using the z-score formula:
\[
z = \frac{X - \mu}{\sigma}
\]

For $ X = 9100 $:
\[
z_1 = \frac{9100 - 25250}{10600}
\]

For $ X = 22850 $:
\[
z_2 = \frac{22850 - 25250}{10600}
\]

After finding these z-scores, use the standard normal distribution table to find the probability corresponding to these z-scores and calculate the difference between them.

### Part c: Middle 10% of Loan Debt
This part asks for the loan debt values that correspond to the middle 10% of the distribution. To solve this, you need to find the z-scores that correspond to the 5th percentile and the 95th percentile (since the middle 10% leaves 5% on each side of the distribution).

- Find the z-scores for 5th and 95th percentiles using a standard normal table.
- Convert the z-scores back to the X-values using the formula:
\[
X = \mu + z \times \sigma
\]

---

Would you like help calculating the specific values for part b and c, or any further clarifications on the steps above? 

Here are five related questions that build upon this:
1. How do you calculate a z-score for a given value?
2. What is the significance of the standard normal distribution table?
3. How does the area under the normal distribution curve relate to probabilities?
4. Why does the middle 10% correspond to the 5th and 95th percentiles?
5. How does changing the standard deviation affect the shape of a normal distribution?

**Tip**: Always check if a problem requires rounding, especially in statistics, to avoid rounding errors in your final answer.

The average student loan debt for college graduates is $25,250. Suppose that the distribution is normal and that the standard deviation is $10,600. Let X = the student loan debt of a randomly selected college graduate. Round all probabilities to 4 decimal places and all dollar answers to the nearest dollar.

Part a: What is the distribution of X?
Part b: Find the probability that the college graduate has between $9,100 and $22,850 in student loan debt.
Part c: The middle 10% of college graduates' loan debt lies between what two numbers?

Solve a probability problem involving the normal distribution of student loan debt. This example calculates the probability that a college graduate has between $9,100 and $22,850 in loan debt and finds the middle 10% loan debt range. Detailed step-by-step solutions are provided using z-scores and the standard normal distribution.

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