If the grades are normally distributed with mean % of Marks 75 & S .D 10 , determine the lowest % of marks to receive grade A & the lowest % Marks that passes.

Subject: Applied Mathematics 4

Topic: Probability

Difficulty: Medium

Let X$_1$ be the minimum marks for which the student will be given grade A

$ P(X \leq X_1) = \frac{15}{100} = 0.15 \\ P(X \leq X_2) = \frac{20}{100} = 0.20 \\ \mu = 75; \hspace{0.5cm} \sigma = 10 $

When, $ X = X_1, \\ Z_1 = \frac{X_1 - 75}{10} \\ $

$ P(Z \geq Z_1) = 0.15 \\ P(0 \leq Z \leq Z_1) = 0.35 \\ \therefore Z_1 = 1.04 \\ \therefore 1.04 = \frac{X_1 - 75}{10} \implies X_1 = 85.4 $

Therefore the minimum % marks at which grade A is given to the student is 85.4

When, $ X = X_2, \\ Z_2 = \frac{X_2 - 75}{10} \\ $

$ P(X \leq X_2) = 0.20 \\ P(Z_2 \leq Z \leq 0) = 0.30 \\ P(0 \leq Z \leq Z_2) = 0.30 \hspace{0.50cm} [Symmetry] \\ Z_2 = -0.85 \\ \therefore \frac{X_2 - 75}{10} = -0.85 \implies X_2 = 66.5 $

Therefore The lowest % that passes is 66.5.