Question

At one college, GPA's are normally distributed with a mean of 2.9 and a standard deviation of 0.6. Using the empirical rule, what percentage of students at the college have a GPA between 2.3 and 3.5?
1)95%
2)84.13%
3)99.7%
4)68%

Answer 1

To use the empirical rule, we first need to determine how many standard deviations the given GPA range (2.3 to 3.5) is from the mean.

1. **Calculate the z-scores**:
- Mean (\( \mu \)) = 2.9
- Standard deviation (\( \sigma \)) = 0.6

For the lower bound (2.3):
\[
z = \frac{(2.3 - 2.9)}{0.6} = \frac{-0.6}{0.6} = -1
\]

For the upper bound (3.5):
\[
z = \frac{(3.5 - 2.9)}{0.6} = \frac{0.6}{0.6} = 1
\]

2. **Apply the empirical rule**:
The empirical rule states that:
- Approximately 68% of the data falls within 1 standard deviation (\( \mu \pm 1\sigma \)) of the mean.
- Approximately 95% falls within 2 standard deviations (\( \mu \pm 2\sigma \)).
- Approximately 99.7% falls within 3 standard deviations (\( \mu \pm 3\sigma \)).

Since 2.3 and 3.5 are 1 standard deviation below and above the mean, respectively, the range [2.3, 3.5] encompasses about 68% of the students.

Therefore, the percentage of students at the college who have a GPA between 2.3 and 3.5 is **68%**.

The correct answer is **4) 68%**.