Interactive Exercise #1 Regression with a small dataset

a. Correlation, Slope, and Y-intercept

r = .535, b = 2.0, a = 2.0

b. The regression equation is y = a +bx . a = y-intercept, b = Slope, y  = predicted score for x.

Case

X

Y

Y

1

1

2

2.0+2.0(1) = 4

2

1

6

2.0+2.0(1) = 4

3

2

5

2.0+2.0(2) = 6

4

2

7

2.0+2.0(2) = 6

Sum

 

 

 

c. SS Total Calculations

Case

X

Y

Y

1

1

2

4

2-5 = -3

(-3)2 = 9

2

1

6

4

6-5 = 1

(1)2 = 1

3

2

5

6

5-5 = 0

(0)2 = 0

4

2

7

6

7-5 = 2

(2)2 = 4

Sum

 

 

 

 

Σ = 14

d. Largest deviation from mean is 3 for Case 1.

e. Contribution of SS Total for Case 3 is 0 because Y = 5 is equal to the mean.

f. See answer for c above.

g. SS Error Calculations  

Case

X

Y

Y

(Y - Y)

(Y - Y)2

1

1

2

4

2-4 = -2

(-2)2 = 4

2

1

6

4

6-4 =  2

(2)2 = 4

3

2

5

6

5-6 = -1

(-1)2 = 1

4

2

7

6

7-6 =  1

(1)2 = 1

Sum

 

 

 

 

Σ = 10

h. The largest deviations are for Cases 1 and 2, and the size of the deviation is 2.The smallest deviations are for Cases 3 and 4, and the size of the deviation is 1.

i. See g above. Both values should be 10

j. SS Predicted Calculations

Case

X

Y

Y

1

1

2

4

4-5 = -1

(-1)2 = 1

2

1

6

4

4-5 = -1

(-1)2 = 1

3

2

5

6

6-5 = 1

(1)2 = 1

4

2

7

6

6-5 = 1

(1)2 = 1

Sum

 

 

 

 

Σ = 4

l. See j above

o. r-squared as proportion of variance explained

[SS Predicted/ SS Total] = 4 / 14 = .286.

Applet reports r = .535; r squared = .286

Back to Module 4 Exercise 1