Question

The result of the nationwide aptitude test in mathematics are normally distributed with a mean of 70 and standard deviation of 5. Find the raw score such that 60% of the cases are below it and case above it.  

Answer 1

Let X be a random variable representing test result, then X ~ $N(70,5^2)$

Let a be such value that 60% scores are below it, then

$P(X<a)=0.6\implies P(N(70,5^2)<a)=0.6\implies P(70+5N(0,1)<a)=0.6\implies P(N(0,1)<{\frac {a-70} 5})=0.6\implies {\frac {a-70} 5}=0.254\implies a=71.27$

Let b be such value that 60% scores are above it, then, due to symmetric of the normal distribution

$b=70-(a-70)=140-71.27=68.73$