Question

According to the College Board, scores on the math section of the SAT Reasoning college entrance test in a recent year had a mean of 511 and a standard deviation of 119. Assume that they are roughly normal.

(a) What was the interval spanned by the middle 95% of SAT scores?

(b) How high must a student score to be in the top 16% of SAT scores?

Answer 1

solution:

middle 95% of score is

P(-z < Z < z) = 0.95

P(Z < z) - P(Z < -z) = 0.95

2 P(Z < z) - 1 = 0.95

2 P(Z < z) = 1 + 0. 95= 1.95

P(Z < z) = 1.95/ 2 = 0.975

P(Z <1.96 ) = 0.975

z  ±1.96

Using z-score formula  

x= z * +

x= ±1.96*119+511

x= 277.76 , 744.24

interval between 277.76 to 744.24

b.

P(Z > z) =16%

= 1 - P(Z < z) = 0.16

= P(Z < z ) = 1 - 0.16

= P(Z < z ) = 0.84

= P(Z < 0.99 ) = 0.84

z =0.99 (using standard normal (Z) table )

Using z-score formula  

x = z * +

x= 0.99*119+511

x= 628.81

x =623 rounded