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.