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This problem involves a normal distribution with a mean of 52 and a standard deviation of 13. Here's how we can solve each part:

### (a) Approximately 68% of the students scored between [ ] and [ ].
In a normal distribution, approximately 68% of the data lies within one standard deviation of the mean. So we need to calculate the range:

- The lower limit is the mean minus one standard deviation: \( 52 - 13 = 39 \).
- The upper limit is the mean plus one standard deviation: \( 52 + 13 = 65 \).

Thus, 68% of the students scored between **39 and 65**.

### (b) Approximately [ ]% of the students scored between 13 and 91.
To solve this, we need to figure out how many standard deviations 13 and 91 are from the mean of 52.

- For a score of 13: \( \frac{13 - 52}{13} = -3 \) standard deviations below the mean.
- For a score of 91: \( \frac{91 - 52}{13} = 3 \) standard deviations above the mean.

In a normal distribution, about **99.7%** of the data lies within 3 standard deviations of the mean. Therefore, approximately **99.7%** of the students scored between 13 and 91.

Would you like further clarification or details? Here are some follow-up questions to explore:

1. What percentage of the students scored below 39?
2. What percentage of the students scored above 65?
3. What is the range that covers 95% of the students’ scores?
4. How many standard deviations from the mean cover 95% of the data?
5. Can you explain the empirical rule for normal distributions?

**Tip**: The empirical rule (68-95-99.7) is very useful when analyzing normally distributed data.

On a nationwide test taken by high school students, the mean score was 52 and the standard deviation was 13. Complete the following statements: (a) Approximately 68% of the students scored between [ ] and [ ]. (b) Approximately [ ]% of the students scored between 13 and 91.

Solve problems involving a nationwide test where the mean score was 52 and the standard deviation was 13. This example uses the empirical rule for normal distributions, helping students learn how to calculate score ranges for 68% and 99.7% of the data. Suitable for high school students in Grades 9-12.

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