Final answer:
To find the percentage of students with a GPA between 1.9 and 3.9, we need to calculate the proportion of students within this range. Let's calculate the z-scores for the lower and upper bounds of the range and use a standard normal distribution table to find the corresponding percentages. The percentage of students with a GPA between 1.9 and 3.9 is approximately 97.59%.
Step-by-step explanation:
To find the percentage of students with a GPA between 1.9 and 3.9, we need to calculate the proportion of students within this range.
First, we convert the lower and upper bounds of the range to z-scores using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
Once we have the z-scores, we can use a standard normal distribution table to find the corresponding percentages. The percentage is then calculated by subtracting the lower percentage from the upper percentage.
Let's calculate the z-scores:
z1 = (1.9 - 3.1) / 0.4 = -3
z2 = (3.9 - 3.1) / 0.4 = 2
Using the standard normal distribution table, we find that approximately 0.0013 (or 0.13%) of students have a GPA below 1.9, and approximately 0.9772 (or 97.72%) of students have a GPA below 3.9.
The percentage of students with a GPA between 1.9 and 3.9 is:
0.9772 - 0.0013 = 0.9759
So, approximately 97.59% of students have a GPA between 1.9 and 3.9.