Question

Raw scores on behavioral tests are often transformed for easier comparison. A test of reading ability has mean 55 and standard deviation 5 when given to third graders. Sixth graders have mean score 77 and standard deviation 7 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 xnew = 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 75 on the test. Find David's transformed score.


Nancy is a sixth-grade student who scores 75. What is her transformed score?


Who scores higher within his or her grade?

Answer 1

(a)

Third - grade scores:

Mean = = 55

SD = s_x = 5

y = a + bx

mean of y = a + (b X mean of x)

100 =a + 55b                       (1)

Variance of y = b² X Variance of x

400 = b² X 25                     (2)

From (2), we get:

b = 4

From (1),

a = 100 - (55 X 4) = -120

So,

a = - 120

b = 4

So,

Answer is:

a =	-120
b =	4

(b)

Sixth - grade scores:

Mean = = 77

SD = s_x = 7

y = a + bx

mean of y = a + (b X mean of x)

100 =a + 77b                       (3)

Variance of y = b² X Variance of x

400 = b² X 49                     (4)

From (4), we get:

b = 2.8571

From (1),

a = 100 - (77 X 2.8571) = -120

So,

a = - 120

b = 2.8571

So,

Answer is:

a =	-120
b =	2.8571

(c)

David:

Third Grade:

For x = 75:

Nancy:

Sixth grade:

For x = 75:

So,

David is higher within his grade.