Question

Please help I am losing the will to live!!
Raw scores on behavioral tests are often transformed for easier comparison. A test of reading ability has mean 51 and standard deviation 10 when given to third graders. Sixth graders have mean score 83 and standard deviation 11 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?

Answer 1

Focus on the third grade class. We have
mean = 51
standard deviation = 10

Answer 2

we want the following
mean = 100
standard deviation = 20

Answer 3

so we want the 51 to turn into 100
we also want the 10 to turn into 20

we essentially have 2 points on a line:
(51,100) and (10,20)

Answer 4

are you able to find the equation of the line through those two points?

Answer 5

I cant remember the formula for that but I thought we had to use the Xnew=a+bx?

Answer 6

do you know how to find the slope of a line using the slope formula?

does this formula look familiar?

Slope Formula
\[\Large m = \frac{y_2 - y_1}{x_2 - x_1}\]

Answer 7

Yes I remember how to do that.
\[m=\frac{ 20-100 }{ 10-51 }=\frac{ -80 }{ -41 }=\frac{ 80 }{ 41 }\]

Answer 8

correct

Answer 9

now let's pick one of the two points, let's say (10,20)

this means x = 10 and y = 20

Answer 10

we'll plug these three values
m = 80/41
x = 10
y = 20

into this slope intercept formula
y = mx+b
then solve for b

Answer 11

\[\Large y = mx+b\]
\[\Large 20 = \frac{80}{41}*10+b\]
solve for b

Answer 12

That would come out to \[\frac{ 20 }{ 41 }\]

Answer 13

yes

Answer 14

m = 80/41
b = 20/41

so the equation y = mx+b turns into
\[\Large y = \frac{80}{41}x+\frac{20}{41}\]

Answer 15

Therefore,
\[\Large X_{\text{new}} = a+b*x\]
turns into
\[\Large X_{\text{new}} = \frac{20}{41}+\frac{80}{41}*x\]

Answer 16

Im trying to understand everything and i feel really confused on how we went from slope formula to this

Answer 17

well y = mx+b and a+bx are the same idea

the 'm' in the first equation and the 'b' in the second equation are the slope. It's unfortunate that 'b' is used in two different ways here. I can see how it's confusing