Raw scores on behavioral tests are often transformed for easier comparison. A te

Question

Raw scores on behavioral tests are often transformed for easier comparison. A test of reading ability has mean 63 and standard deviation 10 when given to third graders. Sixth graders have mean score 81 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 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 72 on the test. Find David's transformed score. Nancy is a sixth-grade student who scores 72. What is her transformed score? Who scores higher within his or her grade? Nancy David

Explanation / Answer

X_new = a + b x

E(X_new) = a + b *E(X)

sd (X_new) = b* sd(X)

E(X_new) = 100 , sd(X_new) = 20

for 3rd grader

    100 = a + b*63

20 = b*10

hence b = 2

a = 100 -2 *63 = -26

X_new = -26 + 2 *x

b) for 6th grader

100 = a + b*81

20 = b*7

hence b = 20/7

a = 100 -20/7 *81 = -131.4285

X_new = -131.4285 + 20/7 *x

c) David

X_new = -26 + 2 *x = -26 +2*72 =118

Nancy

X_new = -131.4285 + 20/7 *x = -131.4285 + 20/7 *72 =   74.28578

Clearly David scored higher as 118 > 74.285

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