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 74 and standard deviation 5 when given to third graders. Sixth graders have mean score 87 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.) 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. Do the same for the sixth-grade scores. David is a third-grade student who scores 78 on the test. Find David's transformed score. Nancy is a sixth-grade student who scores 78. What is her transformed score? Who scores higher within his or her grade?

Explanation / Answer

a) here std deviation =bx

20=b*5

b=4

also E(Xnew) =a+4*E(X)

100=a+4*74

a=-196

b) here std deviaiton 20=7*b

b=20/7=2.8571

hence mean E(Xnew)=a+b*E(X)

100=a+2.8571*87

a=-148.5714

c) David score =-196+4*78 =116

nancy Score =-148.5714+2.8571*78=74.2857

d) David score is higher

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