Question

A large number of applicants for admission to graduate study in business are given an aptitude test. Scores are normally distributed with a mean of 460 and standard deviation of 80. What fraction of the applicants would you expect to have a score of 400 or above?

Answer 1

1)

If X is aptitude test score, then

$P(X\geq 400)=P(\frac{X-\mu }{\sigma }\geq \frac{400-460}{80})=P(Z\geq -0.75)$

$P(Z\geq -0.75)=P(-0.75<Z< 0)+P(Z> 0)$

Refer to standard normal distribution tables, we get

$P(Z\geq -0.75)=0.2734+0.5=0.7734$

Fraction of applicants who score above 400 and above= 0.7734 or 77.34%

2)

$P(Z\leq -1.37)=P(Z\leq 0)-P(-1.37\leq Z<0)$

Refer to standard normal distribution tables, we get

$P(Z\leq -1.37)=0.5-0.4147=0.0853$