The national average for mathematics SATs in 2011 was 514. Suppose that the distribution of scores was approximately bell-shaped and that the standard deviation was approximately 40. Within what boundaries would you expect 68% of the scores to fall? What percentage of scores would be above 594?

We are given that the SAT scores follow a bell-shaped (normal) distribution with:

- Mean (μ) = 514
- Standard deviation (σ) = 40

### Part 1: Boundaries for 68% of the Scores
According to the **Empirical Rule** for normal distributions:
- 68% of the data falls within one standard deviation of the mean, which means between \(\mu - \sigma\) and \(\mu + \sigma\).

Thus, the boundaries are:
\[
\mu - \sigma = 514 - 40 = 474
\]
\[
\mu + \sigma = 514 + 40 = 554
\]

Therefore, 68% of the SAT scores would fall between **474 and 554**.

---

### Part 2: Percentage of Scores Above 594
First, we calculate how many standard deviations 594 is from the mean. This is done using the **z-score** formula:
\[
z = \frac{x - \mu}{\sigma}
\]
Where:
- \(x = 594\)
- \(\mu = 514\)
- \(\sigma = 40\)

Substitute the values:
\[
z = \frac{594 - 514}{40} = \frac{80}{40} = 2
\]
So, a score of 594 is 2 standard deviations above the mean.

Using the standard normal distribution table, a z-score of 2 corresponds to approximately **97.72%** of the data lying below this value. Hence, the percentage of scores above 594 is:
\[
100\% - 97.72\% = 2.28\%
\]

Thus, approximately **2.28%** of the scores would be above 594.

---

Let me know if you would like further details or have any questions.

Here are 5 related questions to explore:
1. What percentage of scores would be below 434?
2. What percentage of scores would fall between 474 and 594?
3. How would the boundaries change if the standard deviation were 50 instead of 40?
4. What is the z-score for a score of 454?
5. What is the probability of getting a score below 474?

**Tip:** The empirical rule is a quick way to estimate percentages in a normal distribution, but for more precision, use z-scores and normal distribution tables.

Learn how to calculate the boundaries for 68% of SAT scores based on the national average in 2011 using the empirical rule. This problem covers key statistical concepts, including the normal distribution, standard deviation, and z-scores, and is suitable for students in Grades 10-12.

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