Answer: (A) To find the percentage of students with a GPA between 2.3 and 3.8, we need to find the area under the normal distribution curve between the corresponding z-scores. First, we find the z-scores for the two GPAs:
z1 = (2.3 - 2.8) / 0.4 = -1.25
z2 = (3.8 - 2.8) / 0.4 = 2.5
Using a standard normal distribution table or calculator, we can find the area to the left of each z-score:
P(Z < -1.25) = 0.1056
P(Z < 2.5) = 0.9938
Then, we can find the area between the two z-scores by taking the difference:
P(-1.25 < Z < 2.5) = 0.9938 - 0.1056 = 0.8882
Therefore, approximately 88.82% of students have a GPA between 2.3 and 3.8.
(B) To find the percentage of students with a GPA of 3.8 or higher, we need to find the area under the normal distribution curve to the right of the corresponding z-score:
z = (3.8 - 2.8) / 0.4 = 2.5
P(Z > 2.5) = 1 - P(Z < 2.5) = 1 - 0.9938 = 0.0062
Therefore, approximately 0.62% of students have a GPA of 3.8 or higher.
(C) Since we know that a student with a GPA of 3.8 or higher makes it on to the Dean's list, we can use the same z-score to find the percentage of students with a GPA of 3.8 or higher:
P(Z > 2.5) = 0.0062
This represents the percentage of students on the Dean's list. To find the total number of students, we can set up a proportion:
0.0062 = 93 / N
Solving for N, we get:
N = 93 / 0.0062 ≈ 15000
Therefore, there are approximately 15000 students at Eastern University.
(D) To find the percentage of students with a GPA below 1.0, we need to find the area under the normal distribution curve to the left of the corresponding z-score:
z = (1.0 - 2.8) / 0.4 = -4.5
Using a standard normal distribution table or calculator, we can find:
P(Z < -4.5) ≈ 0
Therefore, approximately 0% of students have a GPA below 1.0.
Explanation: