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  52 and standard deviation 10 when given to third graders. Sixth graders have mean score 86 and standard deviation 9 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.)

b=

(b) Do the same for the sixth-grade scores.

b=

Explanation / Answer

a) mean = 52 ,sd = 10

mean = 100 and sd = 20

sd(xnew) = sd( a + bx)

20 = b *10 , hence b = 2

mean(xnew) = a + b * mean (x)

= a + 2 * 52

100 = a + 104

hence a= -4

xnew = -4 + 2 *x

b)

similarly

b = 20/9 =2.2222

a = 100 - 20/9 * 86

= -91.111111

xnew = -91.1111 + 2.2222 * x

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