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 453 and standard deviation 103. Scores Y of children of parents with graduate degrees have mean 562 and standard deviation 101. 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 = 60.59 Correct: Your answer is correct.

b = .97 Correct: Your answer is correct.

c =

d = .99 Correct: Your answer is correct.

Explanation / Answer

We have to find such values of a,b,c andd so that

mean(X) = 453, sd(X) =103, mean(Y) = 562, sd(Y) = 101

sd(a+bX) = 100 and sd(c+dY) =100

we know variance is not affected by change in scale

so, sd(a+bX) = b *sd(X) =100

now b * 103=100

b = 0.97

similarly sd(c+dY) = d*sd(Y) = 100

now c*101=100 so d=0.99

Now mean(a+bX) = a +b mean(X) =500

we have b=0.97 and mean(X) = 453

a+(0.97*453)=500

a = 500-439.41 = 60.59

similarly mean (c+dY) = c+d*mean(Y)=500

c + 0.99*562 =500

c =500-556.38 = -56.38

so the values are a=60.59, b=0.97,c=-56.38 and d=0.99

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