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Let's go through the questions on your exam image one by one:

### Question 1:
#### Describe the probability distribution of \(X\) and state its parameters.
The problem states that the SAT score is normally distributed with:
- Mean \( \mu = 1510 \)
- Standard deviation \( \sigma = 351.4 \)

Thus, \( X \sim N(\mu = 1510, \sigma = 351.4) \).

### Question 2:
#### Find the probability for different SAT score ranges:

##### a) The probability that the SAT score is less than 1720.7 points:
We use the standard normal distribution to find this probability. The formula for converting a raw score \(x\) to a z-score is:
\[
z = \frac{x - \mu}{\sigma}
\]
For \(x = 1720.7\):
\[
z = \frac{1720.7 - 1510}{351.4} \approx 0.599
\]
Using the standard normal table or a calculator, the probability of \(z < 0.599\) is approximately:
\[
P(X < 1720.7) \approx 0.7257
\]
Rounding to four decimal places, the probability is **0.7257**.

##### b) The probability that the SAT score is between 2524.4 and 2528 points:
We calculate the z-scores for both values:
\[
z_{2524.4} = \frac{2524.4 - 1510}{351.4} \approx 2.888
\]
\[
z_{2528} = \frac{2528 - 1510}{351.4} \approx 2.891
\]
Now, find the corresponding probabilities from the standard normal distribution:
\[
P(z_{2524.4}) \approx 0.9981 \quad \text{and} \quad P(z_{2528}) \approx 0.9982
\]
Thus, the probability is:
\[
P(2524.4 < X < 2528) = 0.9982 - 0.9981 = 0.0001
\]
So, the probability is approximately **0.0001**.

##### c) The probability that the SAT score is more than 1809.6 points:
We calculate the z-score for \(x = 1809.6\):
\[
z = \frac{1809.6 - 1510}{351.4} \approx 0.854
\]
Using the standard normal table:
\[
P(X > 1809.6) = 1 - P(z < 0.854) \approx 1 - 0.8023 = 0.1977
\]
So, the probability is approximately **0.1977**.

### Question 3:
#### Find the 95th percentile for the SAT score.
To find the 95th percentile, we look for the z-score that corresponds to a cumulative probability of 0.95. Using the z-table, we get:
\[
z_{0.95} \approx 1.645
\]
Now, we convert the z-score back to the original SAT scale using:
\[
x = \mu + z \cdot \sigma = 1510 + 1.645 \cdot 351.4 \approx 2089.2
\]
Rounding to one decimal place, the 95th percentile is **2089.2 points**.

---

Would you like further explanations or details on any part of this? Here are some related questions for further exploration:

1. What is the formula for calculating z-scores, and why is it important?
2. How can we interpret the probabilities in the context of SAT scores?
3. What is the significance of the 95th percentile in standardized testing?
4. How does changing the mean or standard deviation affect the distribution?
5. How do cumulative probabilities relate to real-world scenarios?

**Tip:** Always double-check calculations using a reliable z-score table or calculator to ensure accuracy when dealing with normal distributions.

Historically, the SAT score is normally distributed with a mean of 1510 points and a standard deviation of 351.4 points. Let X be the SAT score of a randomly selected student. 1. Describe the probability distribution of X and state its parameters μ and σ. 2. Find the probability that the SAT score of a randomly selected student is (a) less than 1720.7 points, (b) between 2524.4 and 2528 points, (c) more than 1809.6 points. 3. Find the 95-th percentile for the SAT score.

Learn how to analyze SAT scores using normal distribution with a mean of 1510 and a standard deviation of 351.4. Calculate probabilities for score ranges and find the 95th percentile using z-scores.

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