Question

Raw scores on behavioral tests are often transformed for easier comparison. A test of reading ability has mean 70 and standard deviation 5 when given to third graders. Sixth graders have mean score 80 and standard deviation 13 on the same test. To provide separate "norms" for each grade, we want scores in each grade to have mean 100 and standard deviation 20. (Round your answers to two decimal places.)

(a) What linear transformation will change third-grade scores x into new scores x_new = a + bx that have the desired mean and standard deviation? (Use b > 0 to preserve the order of the scores.)

a =

b =

(b) Do the same for the sixth-grade scores.

a =

b =

(c) David is a third-grade student who scores 73 on the test. Find David's transformed score.


Nancy is a sixth-grade student who scores 73. What is her transformed score?


Who scores higher within his or her grade?

Nancy David    

Answer 1

The Formula is:

on simplifying it becomes,

y = [ (m2*s1 - m1* s2) / s1] + x * (s2/s1)

So,

a = (m2*s1 - m1* s2) / s1

b = s2/s1

1.)

Here, m1 = 70

s1 = 5

m2 = 100

s2 = 20

So, a = (500 - 1400)/5

a = -180

b = 20/5 = 4

So, Xnew = -180 + 4 *X

2.)

Here, m1 = 80

s1 = 13

m2 = 100

s2 = 20

So, a = (1300 - 1600) / 13

a = -300/13 = -23.077

b = 20/13 = 1.538

3.)

For David,

x = 73

Xnew = -180 + 4 * 73

Xnew = 112

For Nancy,

x = 73

Xnew = -23.077 + 1.538 * 73

Xnew = 89.197