Suppose the distribution of scores on the SAT (a college entrance exam
commonly used in the United States) is approximately normal with a
mean of μ = 500 and a standard deviation of = 100. For the population
of students who have taken the SAT, what proportion have SAT scores less
than 450 or greater than 600?

To solve this problem, we need to find the proportion of SAT scores that are less than 450 or greater than 600. Since the distribution of SAT scores is approximately normal with a mean (\(\mu\)) of 500 and a standard deviation (\(\sigma\)) of 100, we can use the standard normal distribution (z-distribution) to calculate these proportions.

### Step 1: Convert the SAT scores to z-scores

The z-score formula is:
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \(X\) is the score,
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.

#### For \(X = 450\):
\[
z = \frac{450 - 500}{100} = \frac{-50}{100} = -0.5
\]

#### For \(X = 600\):
\[
z = \frac{600 - 500}{100} = \frac{100}{100} = 1.0
\]

### Step 2: Find the corresponding probabilities

Using the standard normal distribution table or a calculator:

- The probability that \(z < -0.5\) (SAT score less than 450) is approximately 0.3085.
- The probability that \(z < 1.0\) (SAT score less than 600) is approximately 0.8413.

### Step 3: Calculate the desired proportion

We need to find:
1. The proportion of students with scores less than 450.
2. The proportion of students with scores greater than 600.

The proportion of students with scores greater than 600 is the complement of the probability that \(z < 1.0\):
\[
P(X > 600) = 1 - P(z < 1.0) = 1 - 0.8413 = 0.1587
\]

The total proportion of students with scores either less than 450 or greater than 600 is:
\[
P(X < 450) + P(X > 600) = 0.3085 + 0.1587 = 0.4672
\]

### Final Answer
The proportion of students who have SAT scores less than 450 or greater than 600 is approximately **0.4672** or **46.72%**.

---

Would you like more details or have any questions about this solution?

Here are 5 related questions you might find interesting:
1. What proportion of students have SAT scores between 450 and 600?
2. How would the proportion change if the standard deviation increased to 150?
3. What is the probability of a student scoring exactly 500 on the SAT?
4. How would you calculate the SAT score corresponding to the 90th percentile?
5. What are the implications of a higher mean SAT score on college admissions?

**Tip:** Understanding how to convert raw scores to z-scores is crucial in interpreting normal distributions. Practice with different values to strengthen your grasp!

Suppose the distribution of scores on the SAT (a college entrance exam commonly used in the United States) is approximately normal with a mean of μ = 500 and a standard deviation of = 100. For the population of students who have taken the SAT, what proportion have SAT scores less than 450 or greater than 600?

Explore how to calculate the proportion of students with SAT scores less than 450 or greater than 600 using the normal distribution with a mean of 500 and a standard deviation of 100. Learn about z-scores, probability calculations, and properties of the normal distribution. Ideal for high school students in Grades 10-12.

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