How Laude? Many educational institutions award three levels of Latin honors ofte

Question

How Laude? Many educational institutions award three levels of Latin honors often based on GPA. These are laude (with high praise), magna laude (with great praise), and summa laude (with highest praise). Requirements vary from school to school. Suppose the GPAs at State College are normally distributed with a mean of 2.8 and standard deviation of 0.49.

(a) Suppose State College awards the top 2% of students (based on GPA) with the summa laude honor. What GPA gets you this honor? Round your answer to 2 decimal places.
GPA or higher

(b) Suppose State College awards the top 10% of students (based on GPA) with the magna laude honor. What GPA gets you this honor? Round your answer to 2 decimal places.
GPA or higher

(c) Suppose State College awards the top 20% of students (based on GPA) with the laude honor. What GPA gets you this honor? Round your answer to 2 decimal places.
GPA or higher

Explanation / Answer

a)P(X > x) = 0.02

or, P((X - mean)/sd > (x - 2.8)/0.49)) = 0.02

or, P(Z > (x - 2.8)/0.49) = 0.02

or, P(Z < (x - 2.8)/0.49)) = 0.98

or, (x - 2.8)/0.49 = 2.05

or, x = 2.05 * 0.49 + 2.8

or, x = 3.80

b) P(X > x) = 0.1

or, P((X - mean)/sd > (x - 2.8)/0.49)) = 0.1

or, P(Z > (x - 2.8)/0.49) = 0.1

or, P(Z < (x - 2.8)/0.49)) = 0.9

or, (x - 2.8)/0.49 = 1.28

or, x = 1.28 * 0.49 + 2.8

or, x = 3.43

c) P(X > x) = 0.2

or, P((X - mean)/sd > (x - 2.8)/0.49)) = 0.2

or, P(Z > (x - 2.8)/0.49) = 0.2

or, P(Z < (x - 2.8)/0.49)) = 0.8

or, (x - 2.8)/0.49 = 0.84

or, x = 0.84 * 0.49 + 2.8

or, x = 3.21

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