At one college, PA's are normally distributed with a mean of 2.7 and a standard deviation of 0.4. What percentage of students at the college have a GPA between 23
and 3.1? Use the 68-95-99.7 rule.
To solve this problem using the 68-95-99.7 rule, we need to find the area under the curve between two values.
First, let's convert the GPA values of 2.3 and 3.1 to standard deviations from the mean.
For 2.3:
Z = (X - μ) / σ
Z = (2.3 - 2.7) / 0.4
Z = -0.4 / 0.4
Z = -1
For 3.1:
Z = (X - μ) / σ
Z = (3.1 - 2.7) / 0.4
Z = 0.4 / 0.4
Z = 1
Next, we check the corresponding areas under the curve for these Z-values in the standard normal distribution table.
From the table:
- The area to the left of Z = -1 is approximately 0.1587.
- The area to the left of Z = 1 is approximately 0.8413.
We can calculate the area between the two Z-values by subtracting the smaller area from the larger area:
Area = 0.8413 - 0.1587
Area = 0.6826
This means that approximately 68.26% of students at the college have a GPA between 2.3 and 3.1.