Among other attributes for membership in a honor society at

Among other attributes for membership in a honor society at a nearby high school, one of the qualifications is that the student scores in the top 25% of the national merit exam. If the mean is 560 and the standard deviations is 40, find the lowest acceptable score required to be eligible for membership.

Solution

Normal Distribution
Mean ( u ) =560
Standard Deviation ( sd )=40
Normal Distribution = Z= X- u / sd ~ N(0,1)                  

P ( Z > x ) = 0.25
Value of z to the cumulative probability of 0.25 from normal table is 0.67
P( x-u/ (s.d) > x - 560/40) = 0.25
That is, ( x - 560/40) = 0.67
--> x = 0.67 * 40+560 = 586.96