On the SAT the average Writing score is 480 points and the a

On the SAT, the average Writing score is 480 points and the average Mathematics score is 555 points. The standard deviation is 90 points on each part of the test. Now suppose 55 test takers are selected.

12% of the time the sample average Mathematics score should be less than what? (Remember the label.)

Please show your work so i can learn this!!!

Solution

Normal Distribution
Mean ( u ) =555
Standard Deviation ( sd )= 90/Sqrt(55) = 12.136
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
P ( Z < x ) = 0.12
Value of z to the cumulative probability of 0.12 from normal table is -1.175
P( x-u/s.d < x - 555/12.136 ) = 0.12
That is, ( x - 555/12.136 ) = -1.17
--> x = -1.17 * 12.136 + 555 = 540.7402 ~ less than 541 score