Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.55 and a standard deviation of 0.43. Using the empirical rule, what percentage of the students have grade point averages that are between 2.12 and 2.98?
To find the percentage of students with grade point averages between 2.12 and 2.98, we can use the empirical rule, also known as the 68-95-99.7 rule.
The empirical rule states that for a bell-shaped distribution:
- Approximately 68% of data falls within one standard deviation of the mean.
- Approximately 95% of data falls within two standard deviations of the mean.
- Approximately 99.7% of data falls within three standard deviations of the mean.
In this case, we know that the mean grade point average is 2.55, and the standard deviation is 0.43.
Step 1: Calculate the z-scores for the lower and upper bounds.
To use the empirical rule, we need to convert the GPA values to z-scores. The z-score formula is given by:
z = (x - μ) / σ
For the lower bound:
z1 = (2.12 - 2.55) / 0.43
For the upper bound:
z2 = (2.98 - 2.55) / 0.43
Step 2: Calculate the percentage using the z-scores.
Using the z-scores, we can estimate the percentage of students within the given range.
The percentage from the lower bound to the upper bound is approximately equal to the percentage between z1 and z2. We can find this percentage by referring to a standard normal distribution table or using a calculator with a built-in normal distribution function.
By finding the respective z-scores in the table or using a calculator, we can determine that the percentage between z1 and z2 is approximately 81.85%.
Therefore, approximately 81.85% of the students have grade point averages between 2.12 and 2.98.