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 451 and standard deviation 101. Scores Y of children of parents with graduate degrees have mean 552 and standard deviation 107. 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.)

Explanation / Answer

We have,

E(X) = 451 E(Y) = 1552

SD(X) = 101 SD(Y) = 107

E(a+bX) = 500 E(c+dY) = 500 both have mean 500

SD(a+bX) = 100 SD(c+dY) = 100

Consider,

E(a+bY) = 500

a+bE(X) = 500 By property of expectation

a + 451 b = 500 ...................(1)

Consider,

SD(a+bX) = 100

bSD(X) = 100

101*b = 100

b = 100/101

b =0.99

Now, equation (1) becomes

a + 451*0.99 = 500

a = 500-446.49

a = 53.51

Similarly,

E(c+dY) = 500

c+ dE(Y) = 500

c+ 552*b = 100 .......................(2)

Consider,

SD(c+dY) = 100

d*SD(Y) = 100

d*107 = 100

d = 100/107

d = 0.9346

equation (2) becomes

c+ 552*b = 100

c+ 552* 0.9346 = 552

c = 552 - 515.8992

c = 36.1008

Hence,

a = 53.51 b = 0.99 c = 36.1008 d = 0.9346

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