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

Question

Ram scores on behavioral tests are often transformed for easier comparison. A test of reading ability has mean 56 and standard deviation 10 when given to third graders. Sixth graders have mean score 78 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 that have been 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 students who scores 70 on the test. Find David's transformed score. Nancy is a sixth-grade student who scores 70. What is her transformed score? Who scores higher within his or her grade?

Explanation / Answer

Solution

Let X = actual score of third grader, Y = actual score of sixth grader, and Z = transformed score.

Then, we should have: Z = a1 + b1X …………………………………………(1)

and Z = a2 + b2Y …………………..…………………………………………(2)

Back-up Theory

If P = a + bQ, then meanP = a + bmeanQ …………………………………….(3)

and SD(P) = b.SD(Q) ………………………………………………….(4)

Part (a)

Given meanX = 56 and SD(X) = 10, by (3) and (4), meanZ = a1 + 56b1 and SD(Z) = 10b1.

We want: meanZ = 100 and SD(Z) = 20. => 10b1 = 20 or b1 = 2 and a1 + (56 x 2) = 100 or a1 = - 12.

So, required transforming factors are: a = - 12 and b = 2. ANSWER

Part (b)

Given meanY = 78 and SD(Y) = 13, by (3) and (4), meanZ = a2 + 78b2 and SD(Z) = 13b2.

We want: meanZ = 100 and SD(Z) = 20. => 13b2 = 20 or b2 = 20/13 and a1 + {78 x (20/13)} = 100 or a2 = - 4.

So, required transforming factors are: a = - 4 and b = 1.54. ANSWER

Part (c)

Since David is a third grader, a = - 12 and b = 2 and hence transformed score

= - 12 + (2 x 70) = 128 ANSWER

Part (d)

Since Nancy is a sixth grader, a = - 4 and b = 1.54 and hence transformed score

= - 4 + (1.54 x 70) = 103.70 ANSWER

Part (e)

From Part (c) and Part (d), transformed score of David > transformed score of Nancy. So,

ANSWER is David

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