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 448 and standard deviation 106. 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.)

Answer 1

(1)

For X:

Mean = 448

SD = 106

Let

Z = a + bX

Given:
Mean of Z = 500

SD of Z = 100

By Theorem:
Mean of Z = a + (b * Mean of X)

and

SD of Z = * SD of X

Substituting values, we get:

500 = a + 448b                        (1)

100 = 106                           (2)

From (2),

Substituting in (1), we get:

So,

a = 101.2816

b = 0.89

(2)

For Y:

Mean = 562

SD = 101

Let

Z = c + dY

Given:
Mean of Z = 500

SD of Z = 100

By Theorem:
Mean of Z = c + (d * Mean of Y)

and

SD of Z = * SD of Y

Substituting values, we get:

500 = a + 562d                        (1)

100 = 101                           (2)

From (2),

Substituting in (1), we get:

So,

c = - 50.9286

d = 0.9803