Question

only the top 15% of applicants are considered for prestige medical school based on thier mcat score. the mean mcat score is 500 and the standard deviation is 14. assume that MCAT score are normally distributed. determine the minimum required MCAT score to be considered for prestige medical school.(i.e. determine the score which separates the lower 85% from the top 15%).

Answer 1

Let X denotes the MCAT score of a randomly selected applicant.

X ~ Normal(500, 14²)

To find top 15% of applicants,i.e. to find x such that

$\Rightarrow 1 - P(X\leq x) = 0.15$

$\Rightarrow 1 - P(\frac{X - 500}{14}\leq \frac{x-500}{14}) = 0.15$

$\Rightarrow 1 - P(Z \leq \frac{x-500}{14}) = 0.15$

$\Rightarrow 1 - \Phi(\frac{x-500}{14}) = 0.15$

$\Rightarrow \Phi(\frac{x-500}{14}) =1 - 0.15 = 0.85$

$\Rightarrow \frac{x-500}{14} =\Phi^{-1}(0.85)$

$\Rightarrow \frac{x-500}{14} = 1.03643$

$\Rightarrow x = 500 + (14*1.03643)$

$\Rightarrow x = 514.51$ (ans)