Solving for a, b, c, d

Question

The College Board finds that the distribution of students' SAT scores depends on the level of education their parents have. Children of parents who did not finish high school have SAT math scores X with mean 446 and standard deviation 101. Scores Y of children of parents with graduate degrees have mean 556 and standard deviation 109. Perhaps we should standardize to a common scale for equity. Find numbers a, b, c, and d such that a + bX and c + dY both have mean 500 and standard deviation 100.

Answer 1

To standardize mean 446, standard deviation 101 to 500, 100:

 

b = 100/101 changes the standard deviation to 100

 

Then, 100/101 * 446 + a = 500

a = 500 - 100/101 * 446

a = 58 42/101 = 58.4158

 

mean 556 and standard deviation 109

d = 100/109

c = 500 - 100/109 * 556 = -10 10/109 = -10.0917