The College Board finds that the distribution of students\' SAT scores depends o

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 456 and standard deviation 101. Scores Y of children of parents with graduate degrees have mean 557 and standard deviation 102. 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. (Round your answers to two decimal places.) a = ? b = ? c= ? d= ?

Explanation / Answer

Given condition E(a+bX) =500 ; E(c+dY) =500 ; SD(a+bX) =100 ; SD(c+dY) =100

E(a+bX) = a+bE(X) =500

a+456b =500 ( X with mean 456 and sd 101)

V(a+bX) =b^2 Var(x) = 100^2

b^2 = 100^2/Var(x) = 100^2/101^2 = 0.9803

b=0.9901

substitute and simplify the equation a+456b =500 to get the value a

a= 48.51

Similarly

E(c+dY) = c+557b =500

and V(c+dY) = d^2 Var(Y) = 100^2

d^2 = 100^2/102^2=0.9612

d=0.9804

substitute in equation c+557b =500

c=-46.08

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