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This graph shows the distribution of SAT scores modeled as a normal distribution with a mean (\(\mu\)) of 1051 and a standard deviation (\(\sigma\)) of 211. The shaded region represents the probability or proportion of SAT scores between 750 and 1000. 

To calculate the probability of a score falling within the range \(750 < x < 1000\), we can find the corresponding \(z\)-scores for 750 and 1000 and then use the standard normal distribution table to determine the area under the curve between these \(z\)-scores.

### Step 1: Find the \(z\)-scores
The \(z\)-score formula is:
\[
z = \frac{X - \mu}{\sigma}
\]
For \(X = 750\):
\[
z_{750} = \frac{750 - 1051}{211} \approx -1.43
\]
For \(X = 1000\):
\[
z_{1000} = \frac{1000 - 1051}{211} \approx -0.24
\]

### Step 2: Find the corresponding probabilities
Using a standard normal table (or calculator):
- The probability for \(z = -1.43\) is approximately 0.0764.
- The probability for \(z = -0.24\) is approximately 0.4052.

### Step 3: Calculate the area between the \(z\)-scores
Subtract the smaller area from the larger area to find the probability within the range:
\[
P(750 < x < 1000) = 0.4052 - 0.0764 = 0.3288
\]

So, the probability that a randomly chosen SAT score is between 750 and 1000 is approximately **0.3288**, or **32.88%**.

Would you like further details or have any questions?

### Related Questions
1. What is the probability of scoring below 750 on the SAT?
2. How would the probability change if the score range was from 800 to 1100?
3. What is the probability of scoring above 1000?
4. If the standard deviation changed to 250, how would the probabilities adjust?
5. How do we interpret the z-scores in this context?

### Tip
In a normal distribution, about 68% of values lie within one standard deviation of the mean, so understanding how z-scores map to probabilities is helpful for quick approximations.

What is the probability of scoring between 750 and 1000 on the SAT?

Explore how to calculate the probability of SAT scores falling between 750 and 1000, given a normal distribution with mean 1051 and standard deviation 211. Learn to find z-scores and use the standard normal distribution for probability calculations. Suitable for high school students.

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