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 458 and standard deviation 106. Scores Y of children of parents with graduate degrees have mean 554 and standard deviation 105. 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) = 458 SD(X) = 106

E(Y) = 554 SD(Y) = 105

Consider,

E(a+bX) = 500 (Given)

a+bE(X) = 500 property of expectation/mean

a+b*458 = 500 ........................(1)

Consider,

SD(a+bX) = 100

bSD(X) = 100

b*106 = 100

b = 100/106 = 0.94

eq(1) becomes

a+458*b = 500

a = 500 - 458*b

a = 500-458*(100/106)

a = 67.92

Consider,

SD(c+dY) = 100

d*SD(Y) = 100

d*105 = 100

d = 100/105 = 0.95

Consider,

E(c+dY) = 500

c+dE(Y) = 500

c+ d*554 = 500

c = 500 - 554*d

c = 500 - 554*(100/105) from (d)

c = -27.62

Hence,

a = 67.92 b = 0.94

c = -27.62 d = 0.95

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