Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.53 and a standard deviation of 0.4. Using the empirical rule, what percentage of the students have grade point averages that are greater than 2.13? Please do not round your answer.

2.13 = µ - 1σ

so check your Z table

To solve this problem using the empirical rule, we need to know the number of standard deviations above the mean that 2.13 is.

First, we calculate the z-score of 2.13 using the formula:

z = (X - μ) / σ

Where:
X = grade point average
μ = mean
σ = standard deviation

In this case:
X = 2.13
μ = 2.53
σ = 0.4

Plugging in these values, we get:

z = (2.13 - 2.53) / 0.4
z = -1 / 0.4
z = -2.5

Now we can use the empirical rule to find the percentage of students with grades above 2.13.

According to the empirical rule, for a bell-shaped distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Nearly all (99.7%) of the data falls within three standard deviations of the mean.

Since 2.13 is 2.5 standard deviations below the mean, we can conclude that less than 95% of the students have grades above 2.13.

Therefore, the percentage of students with grades above 2.13 is less than 95%.

To solve this problem using the empirical rule, we need to know the z-score of the given GPA value, which represents the number of standard deviations away from the mean.

The z-score can be calculated using the formula:
z = (x - μ) / σ

Where:
x is the given GPA value (2.13 in this case)
μ is the mean (2.53)
σ is the standard deviation (0.4)

Substituting the values into the formula, we get:
z = (2.13 - 2.53) / 0.4
z = -1 / 0.4
z = -2.5

The z-score of -2.5 tells us that the GPA value of 2.13 is 2.5 standard deviations below the mean.

Now, according to the empirical rule:

- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.

Since the GPA value of 2.13 is more than two standard deviations below the mean, we can conclude that less than 5% of the students will have a GPA greater than 2.13.

Therefore, the percentage of students with a GPA greater than 2.13 is less than 5%.