The regression equation

 

Question 1

The regression equation is intended to be the “best fitting” straight line for a set of data. What is the criterion for “best fitting”?

 

The best fitting line is determined by the error between the predicted Y values on the line and the actual Y values in the data.  The regression equation is determined by the line with the smallest total squared error.

 

Question 2

A set of n=15 pairs of scores (X and Y values) has a SS= 15 and SP = -60. If the mean for the X values is MX= 6 and the mean for the Y values is My= 12, find the regression equation for predicting Y from the X values.

b = SP/ SSX

b = -60/15 = -4

a = MYbMX

a = 12 – (-4)(6) = 12 + 24 = 36

Ŷ = bX + a

Ŷ = -4X + 36

Question 3

A set of n =25 pairs of scores (X and Y values) produces a regression equation of Ŷ = 3X – 6. Find the predicted Y value for each of the following X scores:

  1. 0

Ŷ = 3(0) – 6 = -6

  1. 1

Ŷ = 3(1) – 6 = -3

  1. 3

Ŷ = 3(3) – 6 = 3

  1. -3

Ŷ = 3(-3) – 6 = -15

Question 4

 

We have to find a and b: First we find b

b = SP/ SSX

SP = ƩXY – ƩXƩY

n

SSX = Ʃ(X – MX)2

 

 

 

 

 

 

  1. We find the numerator of b: SP
X Y XY
1 2 2
4 7 28
3 5 15
2 1 2
5 8 40
3 7 21

ƩX = 18        ƩY = 30          ƩXY = 108

SP = ƩXY – ƩXƩY

n

SP = 108 – (18)(30) = 108-90 = 18

6

  1. We find the denominator of b: SSX

SSX = Ʃ(X – MX)2

MX= ƩX/n = 18/6 =3

 

X X – MX (X – MX)2 Y
1 -2 4 2
4 1 1 7
3 0 0 5
2 -1 1 1
5 2 4 8
3 0 0 7

                                            SSX = Ʃ(X – MX)2= 10

SSX = 10

 

 

  1. Calculate b

b = SP/ SSX

b = 18/10 = 1.8

Now we find a

a = MYbMX

MX= ƩX/n = 18/6 =3

MY= ƩY/n = 30/6 =5

a = 5 – (1.8)(3) = 5- 5.4 =  -0.4

The linear regression equation for predicting Y from X is:

Ŷ = bX +

Ŷ = 1.8X – 0.4   

 

Question 5

We have to find a and b: First we find b

b = SP/ SSX

SP = ƩXY – ƩXƩY

n

SSX = Ʃ(X – MX)2

 

  1. We find the numerator of b: SP
X Y XY
3 8 24
6 4 24
3 5 15
3 5 15
5 3 15

ƩX = 20        ƩY = 25         ƩXY = 93

SP = ƩXY – ƩXƩY

n

SP = 93 – (20)(25) = 93 -100 = -7

5

  1. We find the denominator of b: SSX

SSX = Ʃ(X – MX)2

MX= ƩX/n = 20/5 =4

 

X X – MX (X – MX)2 Y
3 -1 1 8
6 2 4 4
3 -1 1 5
3 -1 1 5
5 1 1 3

                                            SSX = Ʃ(X – MX)2= 8

 

SSX = 8

  1. Calculate b

b = SP/ SSX

b = -7/8 = -.875

Now we find a

a = MYbMX

MX= ƩX/n = 20/5 =4

MY= ƩY/n = 25/5 =5

a = 5 – (-.875)(4) = 5 + 3.5 =  8.5

The linear regression equation for predicting Y from X is:

Ŷ = bX +

Ŷ = -.875X + 8.5

Question 6

  1. Use the data to find the regression equation for predicting memory scores from age

We have to find a and b: First we find b

b = SP/ SSX

SP = ƩXY – ƩXƩY

n

SSX = Ʃ(X – MX)2

 

 

 

 

 

 

  1. We find the numerator of b: SP
Age (X) Memory Score (Y) XY
25 10 250
32 10 320
39 9 351
48 9 432
56 7 392

ƩX = 200       ƩY = 45                       ƩXY = 1745

SP = ƩXY – ƩXƩY

n

SP = 1745 – (200)(45) = 1745- 1800 = -55

5

  1. We find the denominator of b: SSX

SSX = Ʃ(X – MX)2

MX= ƩX/n = 200/5 =40

X X – MX (X – MX)2 Y
25 -15 225 10
32 -8 64 10
39 -1 1 9
48 8 64 9
56 16 256 7

                                            SSX = Ʃ(X – MX)2= 610

 

SSX = 610

  1. Calculate b

b = SP/ SSX

b = –55/610 = -.09

 

Now we find a

a = MYbMX

MX= ƩX/n = 200/5 =40

MY= ƩY/n = 45/5 =9

a = 9 – (-.09)(40) = 9 + 3.6 = 12.6

The linear regression equation for predicting Y from X is:

Ŷ = bX +

Ŷ = -.09X + 12.6

Questions 7 to 9

Use the regression equation you found in question 6 to find the predicted memory scores for the following ages: 28, 43, and 50

Ŷ = -0.09(28) + 12.6 = 10.08

Ŷ = -0.09(43) + 12.6 = 8.73

Ŷ = -0.09(50) + 12.6 = 8.1

 

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