Correlation and Linear Regression

Year Percent Used Marijuana

1987 50.20

1988 47.20

1990 40.70

1991 36.70

1992 32.60

1993 35.30

1994 38.2

1995 41.70

1996 44.90

Denomination Total circulation ($)

$1 6253758057

$2 548577377

$5 1468874833

$10 1338391336

$20 4093739605

$50 932552370

$100 2640194345

State Percentage Taking SAT Mean Math SAT

Alaska 48 517

Arizona 29 522

California 45 514

Colorado 30 539

Hawaii 54 512

Idaho 15 539

Montana 22 548

Nevada 32 509

New Mexico 12 545

Oregon 50 524

Utah 4 570

Washington 46 523

Wyoming 12 543

Type Grams of fat (X) Calories (Y)

Hamburger 10 270

Cheeseburger 14 320

Quarter Pounder 21 430

Quarter Pounder w/Cheese 30 530

Big Mac 28 530

10.00 15.00 20.00 25.00 30.00

fat

250.00

300.00

350.00

400.00

450.00

500.00

550.00

calo

ries

Relationship of Fat and Calories in McDonald's Burgers

0.00 10.00 20.00 30.00 40.00 50.00 60.00

Percentage Taking SAT

500.00

510.00

520.00

530.00

540.00

550.00

560.00

570.00

Mea

n M

ath

SA

T

Relationship of Math SAT and Percent Taking Exam

0 20 40 60 80 100

Denomination

0E0

1E9

2E9

3E9

4E9

5E9

6E9

7E9

To

tal C

ircu

lati

on

($

)

Value and Total Circulation of U.S. Currency

1986 1988 1990 1992 1994 1996

Year

35.00

40.00

45.00

50.00

Pe

rcen

t U

se

d M

ari

jua

na

Year of Twelfth Graders and Percentage Who Have Smoked

• How do we measure the strength of a linear relationship? Correlation (r)

• Correlation is a value between

• no correlation

• weak correlation

• moderate correlation

• strong correlation

1 1r

0r

.1r

.3r

.5r

2 2 2 2

XY NXYr

X NX Y NY

Find correlation of McDonald’s fat/calories using above formula.

Type of Burger

Calories (x)

Fat (y)

Hamburger 270 10

Cheeseburger 320 14

Quarter Pounder

430 21

Quarter Pounder w/Chees

e

530 30

Big Mac 530 28

Type Grams of fat (X)

Calories (Y)

XY

X2

Y2

Hamburger 10 270

Cheeseburger 14 320

Quarter Pounder 21 430

Quarter Pounder w/Cheese

30 530

Big Mac 28 530

2 2 2 2

XY NXYr

X NX Y NY

Find correlation of McDonald’s fat/calories using above formula.

Type of Burger

Calories Fat

Hamburger 270 10

Cheeseburger 320 14

Quarter Pounder

430 21

Quarter Pounder w/Chees

e

530 30

Big Mac 530 28

Type Grams of fat (X)

Calories (Y)

XY

X2

Y2

Hamburger 10 270 2700 100 72900Cheeseburger 14 320

Quarter Pounder 21 430

Quarter Pounder w/Cheese

30 530

Big Mac 28 530

2 2 2 2

XY NXYr

X NX Y NY

Find correlation of McDonald’s fat/calories using above formula.

Type of Burger

Calories (x)

Fat (y)

Hamburger 270 10

Cheeseburger 320 14

Quarter Pounder

430 21

Quarter Pounder w/Chees

e

530 30

Big Mac 530 28

Type Grams of fat (X)

Calories (Y)

XY

X2

Y2

Hamburger 10 270 2700 100 72900Cheeseburger 14 320 4480 196 102400

Quarter Pounder 21 430 9030 441 184900

Quarter Pounder w/Cheese

30 530 15900 900 280900

Big Mac 28 530 14840 784 280900

2 2 2 2

XY NXYr

X NX Y NY

Find correlation of McDonald’s fat/calories using above formula.

Type of Burger

Calories (x)

Fat (y)

Hamburger 270 10

Cheeseburger 320 14

Quarter Pounder

430 21

Quarter Pounder w/Chees

e

530 30

Big Mac 530 28

Type Grams of fat (X)

Calories (Y)

XY

X2

Y2

Hamburger 10 270 2700 100 72900Cheeseburger 14 320 4480 196 102400

Quarter Pounder 21 430 9030 441 184900

Quarter Pounder w/Cheese

30 530 15900 900 280900

Big Mac 28 530 14840 784 280900

Mean = 20.6

Mean= 416

46950 2421 922,000

2 2 2 2

XY NXYr

X NX Y NY

Let us take a closer look at these variables

Find correlation of McDonald’s fat/calories using above formula.

Type of Burger

Calories (x)

Fat (y)

Hamburger 270 10

Cheeseburger 320 14

Quarter Pounder

430 21

Quarter Pounder w/Chees

e

530 30

Big Mac 530 28

Type Grams of fat (X)

Calories (Y)

XY

X2

Y2

Hamburger 10 270 2700 100 72900Cheeseburger 14 320 4480 196 102400

Quarter Pounder 21 430 9030 441 184900

Quarter Pounder w/Cheese

30 530 15900 900 280900

Big Mac 28 530 14840 784 280900

Mean = 20.6

Mean= 416

46950 2421 922,000

2 2 2 2

46950 [42848]

2421 2121.8 922000 865280

XY NXYr

X NX Y NY

Find correlation of McDonald’s fat/calories using above formula.

Type of Burger

Calories (x)

Fat (y)

Hamburger 270 10

Cheeseburger 320 14

Quarter Pounder

430 21

Quarter Pounder w/Chees

e

530 30

Big Mac 530 28

Type Grams of fat (X)

Calories (Y)

XY

X2

Y2

Hamburger 10 270 2700 100 72900Cheeseburger 14 320 4480 196 102400

Quarter Pounder 21 430 9030 441 184900

Quarter Pounder w/Cheese

30 530 15900 900 280900

Big Mac 28 530 14840 784 280900

Mean = 20.6

Mean= 416

46950 2421 922,000

2 2 2 2

46950 [42848] 4102

(299.2)(56720)2421 2121.8 922000 865280

XY NXYr

X NX Y NY

Find correlation of McDonald’s fat/calories using above formula.

Type of Burger

Calories (x)

Fat (y)

Hamburger 270 10

Cheeseburger 320 14

Quarter Pounder

430 21

Quarter Pounder w/Chees

e

530 30

Big Mac 530 28

Type Grams of fat (X)

Calories (Y)

XY

X2

Y2

Hamburger 10 270 2700 100 72900Cheeseburger 14 320 4480 196 102400

Quarter Pounder 21 430 9030 441 184900

Quarter Pounder w/Cheese

30 530 15900 900 280900

Big Mac 28 530 14840 784 280900

Mean = 20.6

Mean= 416

46950 2421 922,000

2 2 2 2

46950 [42848] 4102 4102

(299.2)(56720) 169706242421 2121.8 922000 865280

XY NXYr

X NX Y NY

Find correlation of McDonald’s fat/calories using above formula.

Type of Burger

Calories (x)

Fat (y)

Hamburger 270 10

Cheeseburger 320 14

Quarter Pounder

430 21

Quarter Pounder w/Chees

e

530 30

Big Mac 530 28

TypeC

Grams of fat (X)

Calories (Y)

XY

X2

Y2

Hamburger 10 270 2700 100 72900Cheeseburger 14 320 4480 196 102400

Quarter Pounder 21 430 9030 441 184900

Quarter Pounder w/Cheese

30 530 15900 900 280900

Big Mac 28 530 14840 784 280900

Mean = 20.6

Mean= 416

46950 2421 922,000

2 2 2 2

46950 [42848] 4102 4102 4102

4119.54(299.2)(56720) 169706242421 2121.8 922000 865280

XY NXYr

X NX Y NY

Find correlation of McDonald’s fat/calories using above formula.

Type of Burger

Calories (x)

Fat (y)

Hamburger 270 10

Cheeseburger 320 14

Quarter Pounder

430 21

Quarter Pounder w/Chees

e

530 30

Big Mac 530 28

Type Grams of fat (X)

Calories (Y)

XY

X2

Y2

Hamburger 10 270 2700 100 72900Cheeseburger 14 320 4480 196 102400

Quarter Pounder 21 430 9030 441 184900

Quarter Pounder w/Cheese

30 530 15900 900 280900

Big Mac 28 530 14840 784 280900

Mean = 20.6

Mean= 416

46950 2421 922,000

2 2 2 2

46950 [42848] 4102 4102 4102.9957

4119.54(299.2)(56720) 169706242421 2121.8 922000 865280

XY NXYr

X NX Y NY

Equation of a Line

2 2

( )XY NXYb

X NX

Y a bX where and 2 2

( )XY NXYb

X NX

a Y bX

Regression Line for McDonald’s Data

46950 42848 410213.71

2421 2121.8 299.2b

a Y bX 416 13.71(20.6) 133.57a

Equation of the line: y = 133.57 + 13.71x

If a new hamburger has 250 calories then it would have ______grams of fat.

Type

Grams of fat (X)

Calories (Y)

XY

X2

Y2

Hamburger 10 270 2700 100 72900Cheeseburger 14 320 4480 196 102400Quarter Pounder 21 430 9030 441 184900Quarter Pounder

w/Cheese30 530 15900 900 280900

Big Mac 28 530 14840 784 280900

Mean = 20.6

Mean= 416

46950 2421 922,000

Equation of a Line

2 2

( )XY NXYb

X NX

Y a bX where and 2 2

( )XY NXYb

X NX

a Y bX

Regression Line for McDonald’s Data

46950 42848 410213.71

2421 2121.8 299.2b

a Y bX 416 13.71(20.6) 133.57a

Equation of the line: y = 133.57 + 13.71x

If a new hamburger has 250 calories then it would have 8.49 grams of fat.