To solve this problem, we'll use the empirical rule, which applies to bell-shaped (normal) distributions. The empirical rule states:

1. Approximately 68% of the data falls within 1 standard deviation of the mean.
2. Approximately 95% of the data falls within 2 standard deviations of the mean.
3. Approximately 99.7% of the data falls within 3 standard deviations of the mean.

Given:

• Mean ($\mu$) = 2.54
• Standard deviation ($\sigma$) = 0.42
• We need to find the percentage of students with GPAs between 1.28 and 3.80.

Step 1: Calculate the number of standard deviations

1. Lower bound: $Z_1 = \frac{1.28 - 2.54}{0.42}$
2. Upper bound: $Z_2 = \frac{3.80 - 2.54}{0.42}$

Let's calculate these:

$Z_1 = \frac{1.28 - 2.54}{0.42} = \frac{-1.26}{0.42} = -3$ $Z_2 = \frac{3.80 - 2.54}{0.42} = \frac{1.26}{0.42} = 3$

Step 2: Apply the empirical rule

• $Z_1 = -3$ and $Z_2 = 3$ represent 3 standard deviations from the mean.
• According to the empirical rule, approximately 99.7% of the data falls within 3 standard deviations of the mean.

Conclusion:

Therefore, approximately 99.7% of the students have GPAs between 1.28 and 3.80.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How does the empirical rule relate to the standard normal distribution?
2. What would the Z-scores be for GPAs of 2.10 and 3.20?
3. How can the empirical rule be applied to non-bell-shaped distributions?
4. What is the probability of a GPA being higher than 3.80 using the empirical rule?
5. How would the percentage change if the standard deviation were smaller?

Tip: When working with the empirical rule, always ensure that the data is approximately normally distributed. This allows the rule to be applied accurately.