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 80 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 79 on the test. Find David's transformed score.


Nancy is a sixth-grade student who scores 79. What is her transformed score?


Who scores higher within his or her grade?

Nancy David    

Answer 1

(a)

For third grades, the test of reading ability has the following parameters:

Mean = = 55

Standard deviation = = 5

Now applying the transformation:

New score has the following desired mean and standard deviation:

Mean = μ = 100

and standard deviation = σ = 20

Therefore:

So,

b = 4

a = -120

b)

For sixth grades, the test of reading ability has the following parameters:

Mean = = 80

Standard deviation = = 7

Now applying the transformation:

New score has the following desired mean and standard deviation:

Mean = μ = 100

and standard deviation = σ = 20

Therefore:

So,

b = 2.86

a = -128.57

c)

David's transformed score at x=79

= 196

Nancy's transformed score at x=79

=97.37

David scored better

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