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 446 and standard deviation 103. Scores Y of children of parents with graduate degrees have mean 555 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.)

a =

b =

c =

d=

Answer 1

Let X is the SAT math scores of children of parents who did not finish high school.

The expected value of X (mean of X ) is

The standard deviation of X is

Now we want to find the values of a and b such that a+bX has a mean 500 and standard deviation 100

We know that for any 2 constants a and b

We want this value to be 500

or

Next we know that for any 2 constants a,b

We want this new standard deviation to be 100

Hence

Hence the value of a is

ans: a=66.99

b=0.97

Let Y is the SAT math scores of children of parents with graduate degrees

The expected value of Y (mean of Y ) is

The standard deviation of Y is

Now we want to find the values of c and d such that c+dY has a mean 500 and standard deviation 100

We know that for any 2 constants c and d

We want this value to be 500

or

Next we know that for any 2 constants c,d

We want this new standard deviation to be 100

Hence

Hence the value of c is

ans: c=-18.69

d=0.93