Let's break down each part of the problem one by one:

Given Information:

• Mean ($\mu$) = 550
• Standard deviation ($\sigma$) = 120
• The scores follow a normal distribution.

A. Probability of a randomly selected student scoring below 450:

We need to find $P(X < 450)$, where $X$ is the score of a randomly selected student.

First, we standardize the score using the z-score formula:

$z = \frac{X - \mu}{\sigma}$

Substituting the values:

$z = \frac{450 - 550}{120} = \frac{-100}{120} = -0.8333$

Now, we need to find the cumulative probability associated with this z-score. You can use a standard normal distribution table, or a calculator to find the probability corresponding to $z = -0.8333$.

Looking up $z = -0.83$ in the standard normal table, the probability is approximately 0.2023. Therefore,

$P(X < 450) = 0.2023$

So, the probability that a randomly selected student scores below 450 is 0.2023 or 20.23%.

---

B. Probability that a student scored between 300 and 400:

We need to find $P(300 < X < 400)$, which can be broken down as:

$P(300 < X < 400) = P(X < 400) - P(X < 300)$

Step 1: Calculate $P(X < 400)$:

$z_{400} = \frac{400 - 550}{120} = \frac{-150}{120} = -1.25$

From the standard normal table, $P(Z < -1.25) \approx 0.1056$.

Step 2: Calculate $P(X < 300)$:

$z_{300} = \frac{300 - 550}{120} = \frac{-250}{120} = -2.0833$

From the standard normal table, $P(Z < -2.08) \approx 0.0188$.

Step 3: Calculate the probability:

$P(300 < X < 400) = P(X < 400) - P(X < 300) = 0.1056 - 0.0188 = 0.0868$

So, the probability that a student scored between 300 and 400 is 0.0868 or 8.68%.

---

C. Probability that a student scored more than 450:

We need to find $P(X > 450)$, which is the complement of the probability $P(X < 450)$.

We already know that $P(X < 450) = 0.2023$, so:

$P(X > 450) = 1 - P(X < 450) = 1 - 0.2023 = 0.7977$

So, the probability that a student scored more than 450 is 0.7977 or 79.77%.

---

Final Answers:

• A. $P(X < 450) = 0.2023$ or 20.23%
• B. $P(300 < X < 400) = 0.0868$ or 8.68%
• C. $P(X > 450) = 0.7977$ or 79.77%

Would you like further clarification on any of these steps or additional examples? Here are some related questions for you to explore:

1. How do we calculate the z-score for other values in a normal distribution?
2. How does changing the mean or standard deviation affect the probabilities in a normal distribution?
3. What are the key differences between normal and other types of distributions, like the binomial distribution?
4. How can we use the z-table more efficiently when calculating probabilities?
5. What is the role of the standard normal distribution in hypothesis testing?

Tip: For any normal distribution problem, remember that the z-score represents how many standard deviations a given value is from the mean. Using this value in a z-table (or calculator) helps determine the cumulative probability.