Assignment 03 - Regression - Need in 14 Hours Max to Max

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2014-09-09RegressionStep-by-Step.docx
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Regression (aka Least Squares)

Regression is the formal name given to using an existing set of data to project what is going to happen in the future. It is also called “least squares” because it creates a line that is the shortest (least) distance from each data point and the line itself.

This is an example of a good regression line that is fit between the various data points on the x-y axes.

Notice how the thick black line fits nicely between all the points.

Regression – Fits a trend line to a series of historical data points and projects the line into the future for medium- to long-range forecasts. It is also called “least squares” because it creates a line that is the shortest (least) distance from each data point and the line itself.

The equation for determining the least squares line is as follows:

(also known as “Y-hat”) equals the predicted position of Y (the dependent variable) on the Y axis.

X equals the position of X (the independent variable) on the X axis.

a and b modify the height of Y based on the position of X.

How to Solve a Regression Problem:

1. Build a data table including all Xs, Ys, XY, X2, and Y2.

2. Sum each column in the data table—that is, X, Y, XY, X2, and Y2.

3. Insert the summed data into the following equation for b:

4. Insert the summed data into the following equation for a:

5. Insert the solutions for a and b into the following equation:

6. Insert the value of X to solve for Y-hat.

Step-by-Step for Regression

Step 1: Build the grid, inserting variables for X and Y.

Cylinders

City MPG

n 

 

X

Y

1

Audi A4 Avant

4

22

2

Audi A8

8

17

3

BMW 325

6

20

4

BMW 525

6

20

5

BMW 645

8

17

6

Dodge Charger

6

19

7

Dodge Magnum

6

21

8

Ford Explorer

6

15

9

Ford Mustang

6

19

10

Honda Civic

4

30

11

Honda Odyssey

6

19

12

Mazda 3

4

28

13

Mercedes Benz E-Class

6

19

14

Nissan Titan

8

14

15

Nissan Xterra

6

16

16

Scion xB

4

30

17

Toyota Tacoma

4

21

Step 2: Using the Xs and Ys, calculate XY, X2, and Y2.

Cylinders

City MPG

 

 

X

Y

XY

X2

Y2

1

Audi A4 Avant

4

22

88

16

484

2

Audi A8

8

17

136

64

289

3

BMW 325

6

20

120

36

400

4

BMW 525

6

20

120

36

400

5

BMW 645

8

17

136

64

289

6

Dodge Charger

6

19

114

36

361

7

Dodge Magnum

6

21

126

36

441

8

Ford Explorer

6

15

90

36

225

9

Ford Mustang

6

19

114

36

361

10

Honda Civic

4

30

120

16

900

11

Honda Odyssey

6

19

114

36

361

12

Mazda 3

4

28

112

16

784

13

Mercedes Benz E-Class

6

19

114

36

361

14

Nissan Titan

8

14

112

64

196

15

Nissan Xterra

6

16

96

36

256

16

Scion xB

4

30

120

16

900

17

Toyota Tacoma

4

21

84

16

441

Step 3: Find the sum of each column.

Cylinders

City MPG

 

 

X

Y

XY

X2

Y2

1

Audi A4 Avant

4

22

88

16

484

2

Audi A8

8

17

136

64

289

3

BMW 325

6

20

120

36

400

4

BMW 525

6

20

120

36

400

5

BMW 645

8

17

136

64

289

6

Dodge Charger

6

19

114

36

361

7

Dodge Magnum

6

21

126

36

441

8

Ford Explorer

6

15

90

36

225

9

Ford Mustang

6

19

114

36

361

10

Honda Civic

4

30

120

16

900

11

Honda Odyssey

6

19

114

36

361

12

Mazda 3

4

28

112

16

784

13

Mercedes Benz E-Class

6

19

114

36

361

14

Nissan Titan

8

14

112

64

196

15

Nissan Xterra

6

16

96

36

256

16

Scion xB

4

30

120

16

900

17

Toyota Tacoma

4

21

84

16

441

Sum

Sum X

Sum Y

Sum XY

Sum X2

Sum Y2

98

347

1916

596

7449

Step 4: Find the average of X and Y.

Cylinders

City MPG

 

 

X

Y

XY

X2

Y2

1

Audi A4 Avant

4

22

88

16

484

2

Audi A8

8

17

136

64

289

3

BMW 325

6

20

120

36

400

4

BMW 525

6

20

120

36

400

5

BMW 645

8

17

136

64

289

6

Dodge Charger

6

19

114

36

361

7

Dodge Magnum

6

21

126

36

441

8

Ford Explorer

6

15

90

36

225

9

Ford Mustang

6

19

114

36

361

10

Honda Civic

4

30

120

16

900

11

Honda Odyssey

6

19

114

36

361

12

Mazda 3

4

28

112

16

784

13

Mercedes Benz E-Class

6

19

114

36

361

14

Nissan Titan

8

14

112

64

196

15

Nissan Xterra

6

16

96

36

256

16

Scion xB

4

30

120

16

900

17

Toyota Tacoma

4

21

84

16

441

Sum

Sum X

Sum Y

Sum XY

Sum X2

Sum Y2

98

347

1916

596

7449

Average

5.76

20.41

Step 5: Solve for b, using the information in the data grid.

Page 3 of 5

Step 6: Solve for a, using the information in the data grid.

Step 7: Put a and b into the regression equation.

Step 8: To solve for Y, insert X.

If X = 6 (cylinders), then what does Y (MPG) equal?

For a 6-cylinder vehicle (X = 6), Y = 19.79 MPG.

^

Y

å

å

-

-

=

2

2

X

n

X

Y

X

n

XY

b

X

b

Y

a

-

=

å

=

98

X

596

2

å

=

X

76

.

5

=

X

å

=

347

Y

å

=

7449

2

Y

41

.

20

=

Y

å

=

1916

XY

17

=

n

å

å

-

-

=

2

2

X

n

X

XY

n

XY

b

(

)

(

)

(

)

2

76

.

5

*

17

596

41

.

20

*

76

.

5

*

17

1916

-

-

=

b

(

)

33.18

*

17

596

1998.55

1916

-

-

=

b

564.06

596

82.55

-

-

=

b

31.94

55

.

82

-

=

b

-2.58

=

b

X

b

Y

a

-

=

(

)

76

.

5

*

58

.

2

41

.

20

-

-

=

a

86

.

14

41

.

20

+

=

a

27

.

35

=

a

bX

a

Y

+

=

ˆ

X

Y

58

.

2

27

.

35

ˆ

-

=

6

*

58

.

2

27

.

35

ˆ

-

=

Y

15.48

27

.

35

ˆ

-

=

Y

79

.

19

ˆ

=

Y

bX

a

Y

+

=

^