Question

A sample of test scores had a mean M = 38 and s = 5. A researcher wants to transform the original test scores into a new data set with a new mean M = 100 and and a new standard deviation s = 10. What would be the new value of the original test score X = 25 after the transformation?

Answer 1

Consider test scores to be normally distributed

X - N(38,25) Y - N(100,100) [25 and 100 are variances of the corresponding standard deviations]

If Z - N(0,1) standard normal

Standard normal of X is X-38/5

therefore, z= x-38/5

5z+38 = x

Let Y be ax +b

y= a (5z+38) +b

y= 5za + (38a+b)

therefore, z = [y- (38a+b)] / 5a ; this is the standard normal conversion of y

this can also be written as [y-100/10]

comparing the two,

38a+b=100 and 5a=10

a=2

b=100-76 = 24

y=2x + 24

y =2*25 +24

= 74 ( new value of x=25)