Busn210ch14 Statistics Series

7/30/2019 Busn210ch14 Statistics Series

1/61

Linear Regression #1: Scatter Diagram: Relationship Between 2 Variables?

Linear Regression #2: Scatter Plot with Trendline & X and Y Mean Lines

#REF!

#REF!

#REF!


2/61


Plotting Two variables: Dont use Line Chart, Use Scatter Chart

Plotting the point on the chart that graphs the relationship between two variables: Move along x axis a give

and then along the y axis a certain amount.

Independent, Predictor Variable = x

Dependent, Predicted Variable = y

Scatter Diagram with proper x and y axis labels to see if there is a relationship between two variabl

Direct, Positive Relationship: As x increases, y increases

Indirect, Negative Relationship: As x increases, y decreases

No relationship: no pattern can be seen

Add Trendline with linear equation and coefficient of determination (goodness of fit: of the total variation,

can model explain?)

Example 1:

Independent Variable Dependent Variable

Predictor Variable Predicted Variable

Sample Point x y

No. Time Studying (hours) Score on Test

1 3 49

2 11 87

3 2 50

4 13 89

5 8 84

6 12 79

7 13 100

8 4 579 7 64

10 14 98

11 7 81

12 7 68

13 14 88

14 4 45

15 4 52

16 5 15

17 12 72

18 16 97

19 12 8920 14 87

21 2 48

22 12 92

23 11 89

24 6 52

25 11 84

26 14 94

27 10 79


3/61


4/61

Example 4:



Sample Point x y

No. Years Using ExcelExpert Level (Rating1 - 10))

1 3 5

2 8 1

3 6 9

4 11 5

5 20 3

6 7 4

7 9 10

8 3 6

9 19 10

10 2 1

11 16 2

12 12 7

13 1 6


5/61

n amount

s.

ow much


6/61


Plotting Two variables: Dont use Line Chart, Use Scatter Chart

Plotting the point on the chart that graphs the relationship between two variables: Move along x axis a give

and then along the y axis a certain amount.

Independent, Predictor Variable = x

Dependent, Predicted Variable = y

Scatter Diagram with proper x and y axis labels to see if there is a relationship between two variabl

Direct, Positive Relationship: As x increases, y increases

Indirect, Negative Relationship: As x increases, y decreases

No relationship: no pattern can be seen

Add Trendline with linear equation and coefficient of determination (goodness of fit: of the total variation,

can model explain?)

Example 1:



Sample Point x y

No. Time Studying (hours) Score on Test

1 3 49

2 11 87

3 2 50

4 13 89

5 8 84

6 12 79

7 13 100

8 4 579 7 64

10 14 98

11 7 81

12 7 68

13 14 88

14 4 45

15 4 52

16 5 15

17 12 72

18 16 97

19 12 8920 14 87

21 2 48

22 12 92

23 11 89

24 6 52

25 11 84

26 14 94

27 10 79

0

20

40

60

80

100

120

0 2 4 6 8 10 12

ScoreonTest

Time Studying (hours)


7/61

28 6 59

29 10 66

30 11 97

Example 2:



Sample Point x y

No. Temperature (F) Sales Chicken Soup

1 86 $3,300

2 40 $8,200

3 41 $8,900

4 78 $3,100

5 71 $4,020

6 91 $1,950

7 70 $2,500

8 37 $6,500

9 65 $6,210

10 42 $5,250

11 53 $7,200

12 83 $2,750

13 63 $7,150

14 36 $7,900

15 43 $6,210

Example 3:



Sample Point x y

No. Temperature (F) Sales Ice Cream

1 91 $7,113

2 45 $2,044

3 46 $1,108

4 83 $7,093

5 76 $3,902

6 96 $6,676

7 75 $5,403

8 42 $886

9 70 $4,740

10 47 $2,637

11 58 $3,150

y = -100.56x + 11436

R = 0.7193

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

$10,000

0 20 40 60

Sa

lesC

hic

kenSoup

Temperature (F)

y = 112x - 3354.1

R = 0.9056

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

0 20 40 60 80

Sa

lesIceCream

Temperature (F)


8/61

Example 4:



Sample Point x y

No. Years Using ExcelExpert Level (Rating1 - 10))

1 3 5

2 8 1

3 6 9

4 11 5

5 20 3

6 7 4

7 9 10

8 3 6

9 19 10

10 2 1

11 16 2

12 12 7

13 1 6

y = 0.0436x + 4.9156

R = 0.0078

0

2

4

6

8

10

12

0 5 10 15 20

ExpertLeve

l(Rating1-

10

))

Years Using Excel


9/61

n amount

s.

ow much

y = 4.2914x + 34.362

R = 0.7266

14 16 18


10/61

80 100

100 120


11/61

5


12/61


1. Create Scatter Plot with Trendline & X and Y Mean Lines to divide chart into four quadrants in order to fur

the pattern and relationship between the two variables

Example 2:

Mean: Xbar y x

Sample Point x y


1 86 $3,300

2 40 $8,200

3 41 $8,900

4 78 $3,100

5 71 $4,020

6 91 $1,950

7 70 $2,5008 37 $6,500

9 65 $6,210

10 42 $5,250

11 53 $7,200

12 83 $2,750

13 63 $7,150

14 36 $7,900

15 43 $6,210

Example 3: Xbar y x66.27273 0 0

Mean: 66.27272727 $4,068 66.27273 8000 120

Sample Point x y


1 91 $7,113

2 45 $2,044

3 46 $1,108

4 83 $7,093

5 76 $3,902

6 96 $6,676

7 75 $5,403

8 42 $886

9 70 $4,740

10 47 $2,637

11 58 $3,150

y = 112x - 3354.1

R = 0.9056

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

0 20 40 60

Sa

les

IceCream

Te

Sales Ice Cream Xbar


13/61

her define

Ybar

Ybar$4,068

$4,068

80 100 120 140

perature (F)

Ybar Linear (Sales Ice Cream)


14/61


1. Create Scatter Plot with Trendline & X and Y Mean Lines to divide chart into four quadrants in order to fur

the pattern and relationship between the two variables

Example 2:

Mean: 59.93333333 $5,409 Xbar y x

59.93333 0 0

Sample Point x y 59.93333 10000 100


1 86 $3,300

2 40 $8,200

3 41 $8,900

4 78 $3,100

5 71 $4,020

6 91 $1,950

7 70 $2,5008 37 $6,500

9 65 $6,210

10 42 $5,250

11 53 $7,200

12 83 $2,750

13 63 $7,150

14 36 $7,900

15 43 $6,210

Example 3: Xbar y x66.27273 0 0

Mean: 66.27272727 $4,068 66.27273 8000 120

Sample Point x y


1 91 $7,113

2 45 $2,044

3 46 $1,108

4 83 $7,093

5 76 $3,902

6 96 $6,676

7 75 $5,403

8 42 $886

9 70 $4,740

10 47 $2,637

11 58 $3,150

y = 112x - 3354.1

R = 0.9056

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

0 20 40 60

Sa

les

IceCream

Te


y = -100.56x + 11436

R = 0.7193

$0

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

0 20 40 60

Sa

lesC

hic

kenSoup

Temperature (F)


15/61

her define

Ybar

$5,409

$5,409

Ybar$4,068

$4,068

80 100 120 140

perature (F)


80 100 120


16/61

Linear Regression #3: Coefficient of Correlation: Strength & Direction of Relationship

Calculate the Sample Covariance long hand to get measure of strength of the linear relationship.

Use Scatter Plot with Trendline & X and Y Mean Lines to see why covariance makes sense

Calculate the Sample Covariance using Excel function COVARIANCE.S

Measure Strength and Direction of Relationship with Coefficient of Correlation

Calculate Coefficient of Correlation long hand to get a measure of the strength and direction of the linear rela

number will vary from -1 to 0 to +1 (minus one to zero to positive one) and will indicate a perfect indirect (

relationship when minus one, no relationship when it is zero and a perfect direct relationship when it is po

Reasonable positive number = Direct, Positive Relationship: As x increases, y increases

Reasonable negative number = Indirect, Negative Relationship: As x increases, y decreases

Number close to zero = No relationship: no pattern can be seen

See three charts to help visualize the three correlation situations.

Calculate Coefficient of Correlation with the Excel functions CORREL and PEARSON

Calculate Sample Standard Deviation long hand to see that it is related to Coefficient of Correlation and other Li

calculations

Xbar y

59.93333333 0

Example 2: 59.93333333 10000

x Ybar

0 $5,409

Mean: 59.93333 $5,409 100 $5,409

Count 15

n -1 14

Sample Point x y (x Deviation) (y Deviation) (x Deviation)^2 (y Deviation)^2

No.

Temperat

ure (F) Sales Chicken Soup (x - Xbar) (y - Ybar) (x - Xbar)^2 (y - Ybar)^2

1 86 $3,300

2 40 $8,200

3 41 $8,900

4 78 $3,100

5 71 $4,020

6 91 $1,950

7 70 $2,500

8 37 $6,500

9 65 $6,21010 42 $5,250

11 53 $7,200

12 83 $2,750

13 63 $7,150

14 36 $7,900

15 43 $6,210

Sum of Deviations

SUM Deviations^2 ====================>>

y = -100.5

R =

$0

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

0 20 40 60 80 1

Sa

lesC

hic

kenSoup

Temperature (F)


17/61

SUM Mult. Deviations =============================================>>

Sample SD x

Sample SD y

Sample Covariance

Coefficient of Correlation

Xbar y x

66.27272727 0 0

Example 3: 66.27272727 8000 120

Sample Point x y

No.

Temperat

ure (F) Sales Ice Cream

1 91 $7,113

2 45 $2,044

3 46 $1,108

4 83 $7,093

5 76 $3,902

6 96 $6,676

7 75 $5,4038 42 $886

9 70 $4,740

10 47 $2,637

11 58 $3,150

Mean: 66.27273 $4,068

Sample

Covariance

Coefficient

of

Correlation Strength and Direction of the relationship

Coefficient of Determination = R^2 = "Goodness of fit for our line" r^2

Example 4:

Sample Point x y

y = 112x - 3354.1

R = 0.9056

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

0 20 40 60 80

Sa

lesIceCream

Temperature (F)

Sales Ice Cream Xbar Ybar Li


18/61

No.

Years

Using

Excel

Expert Level (Rating 1 -

10))

1 3 5

2 8 1

3 6 9

4 11 55 20 3

6 7 4

7 9 10

8 3 6

9 19 10

10 2 1

11 16 2

12 12 7

13 1 6

Mean: 9.636364 5.307692308

Sample

Covariance

Coefficient

of

Correlation

r^2

y = 0.0436

R = 0.

0

2

4

6

8

10

12

0 5 10 1

ExpertLeve

l(Rating1-

10

))

Years Using Excel


19/61

ionship. This

negative)

sitive one.

ear Regression

(x Deviation)*

(y Deviation)

(x Deviation)*

(y Deviation)

6x + 11436

.7193

0 120

Coefficient of Correlation = Measures Strength

and Direction Of Liner Relationship. Does Not

Have A Problem With Units. Range From -1 to

0 to + 1. -1 = Perfect Indirect (Negative)

Relationship (as x increases, y decreases). 0 =

No Relationship. +1 = Perfect Direct (Positive)

Relationship (as x increases, y increases).

Used for Linear Relationship only.

(

)

Sample Standard Deviation = Spread In Data. How

Fairly Does The Mean Represent The Data Points?

s =

2

(1)

Sample Covariance = Measure the Strength of the Linear

Relationship Between 2 Variables, but has problem with units.

Note: See 4 Quadrant Example of why this measure makes

sense.

sxy =

( )

1


20/61

Correlation is not causation

Ybar

$4,068

$4,068

100 120 140

ear (Sales Ice Cream)

xy sxsy


21/61

x + 4.9156

.0078

5 20 25


22/61

Linear Regression #3: Coefficient of Correlation: Strength & Direction of Relationship

Calculate the Sample Covariance long hand to get measure of strength of the linear relationship.

Use Scatter Plot with Trendline & X and Y Mean Lines to see why covariance makes sense

Calculate the Sample Covariance using Excel function COVARIANCE.S

Measure Strength and Direction of Relationship with Coefficient of Correlation

Calculate Coefficient of Correlation long hand to get a measure of the strength and direction of the linear rela

number will vary from -1 to 0 to +1 (minus one to zero to positive one) and will indicate a perfect indirect (

relationship when minus one, no relationship when it is zero and a perfect direct relationship when it is poReasonable positive number = Direct, Positive Relationship: As x increases, y increases

Reasonable negative number = Indirect, Negative Relationship: As x increases, y decreases

Number close to zero = No relationship: no pattern can be seen

See three charts to help visualize the three correlation situations.

Calculate Coefficient of Correlation with the Excel functions CORREL and PEARSON

Calculate Sample Standard Deviation long hand to see that it is related to Coefficient of Correlation and other Li

calculations

Xbar y

59.93333333 0

Example 2: 59.93333333 10000

x Ybar

0 $5,409

Mean: 59.93333 $5,409 100 $5,409

Count 15

n -1 14

Sample Point x y (x Deviation) (y Deviation) (x Deviation)^2 (y Deviation)^2

No.Temperature (F) Sales Chicken Soup (x - Xbar) (y - Ybar) (x - Xbar)^2 (y - Ybar)^2

1 86 $3,300 26.0666667 -2109.33333 679.4711111 4449287.111

2 40 $8,200 -19.9333333 2790.666667 397.3377778 7787820.444

3 41 $8,900 -18.9333333 3490.666667 358.4711111 12184753.78

4 78 $3,100 18.0666667 -2309.33333 326.4044444 5333020.444

5 71 $4,020 11.0666667 -1389.33333 122.4711111 1930247.111

6 91 $1,950 31.0666667 -3459.33333 965.1377778 11966987.11

7 70 $2,500 10.0666667 -2909.33333 101.3377778 8464220.444

8 37 $6,500 -22.9333333 1090.666667 525.9377778 1189553.778

9 65 $6,210 5.06666667 800.6666667 25.67111111 641067.1111

10 42 $5,250 -17.9333333 -159.333333 321.6044444 25387.1111111 53 $7,200 -6.93333333 1790.666667 48.07111111 3206487.111

12 83 $2,750 23.0666667 -2659.33333 532.0711111 7072053.778

13 63 $7,150 3.06666667 1740.666667 9.404444444 3029920.444

14 36 $7,900 -23.9333333 2490.666667 572.8044444 6203420.444

15 43 $6,210 -16.9333333 800.6666667 286.7377778 641067.1111

Sum of Deviations 0.00 0.00

SUM Deviations^2 ====================>> 5272.933333 74125293.33

SUM Mult. Deviations =============================================>>

y = -100.5

R =

$0

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

0 20 40 60 80 1

Sa

lesC

hic

kenSoup

Temperature (F)


23/61

Sample SD x 19.40716608 19.40716608

Sample SD y 2301.013648 2301.013648

Sample Covariance -37874.3333 -37874.33333 -37874.33333

Coefficient of Correlation -0.84813245 -0.84813245

-0.84813245

Xbar y x

66.27272727 0 0

Example 3: 66.27272727 8000 120

Sample Point x y

No.

Temperat


1 91 $7,113

2 45 $2,044

3 46 $1,108

4 83 $7,093

5 76 $3,902

6 96 $6,676

7 75 $5,403

8 42 $8869 70 $4,740

10 47 $2,637

11 58 $3,150

Mean: 66.27273 $4,068

Sample

Covariance 43143.69

Coefficient

of

Correlation 0.951608 Strength and Direction of the relationship

Coefficient of Determination = R^2 = "Goodness of fit for our line" r^2 0.905558201

Example 4:

Sample Point x y

y = 112x - 3354.1

R = 0.9056

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

0 20 40 60 80

Sa

lesIceCream

Temperature (F)

Sales Ice Cream Xbar Ybar Li


24/61

No.

Years

Using

Excel

Expert Level (Rating 1 -

10))

1 3 5

2 8 1

3 6 9

4 11 55 20 3

6 7 4

7 9 10

8 3 6

9 19 10

10 2 1

11 16 2

12 12 7

13 1 6

Mean: 9.636364 5.307692308

Sample

Covariance

Coefficient

of

Correlation 0.088518

r^2 0.007835

y = 0.0436

R = 0.

0

2

4

6

8

10

12

0 5 10 1

ExpertLeve

l(Rating1-

10

))

Years Using Excel


25/61

ionship. This

negative)

sitive one.

ear Regression

(x Deviation)*

(y Deviation)

(x Deviation)*(y Deviation)

-54983.28889

-55627.28889

-66089.95556

-41721.95556

-15375.28889

-107469.9556

-29287.28889

-25012.62222

4056.711111

2857.377778-12415.28889

-61341.95556

5338.044444

-59609.95556

-13557.95556

-530240.6667

6x + 11436

.7193

0 120

Coefficient of Correlation = Measures Strength

and Direction Of Liner Relationship. Does Not

Have A Problem With Units. Range From -1 to

0 to + 1. -1 = Perfect Indirect (Negative)

Relationship (as x increases, y decreases). 0 =

No Relationship. +1 = Perfect Direct (Positive)

Relationship (as x increases, y increases).

Used for Linear Relationship only.

rxy =

(

)

Sample Standard Deviation = Spread In Data. How

Fairly Does The Mean Represent The Data Points?

s =

2

(1)

Sample Covariance = Measure the Strength of the Linear

Relationship Between 2 Variables, but has problem with units.

Note: See 4 Quadrant Example of why this measure makes

sense.

sxy =

( )

1


26/61

Correlation is not causation

Ybar

$4,068

$4,068

100 120 140

ear (Sales Ice Cream)

x


27/61

x + 4.9156

.0078

5 20 25


28/61

Linear Regression #4: Calculate Slope & Y-Intercept, Create Estimated Equation and Use It

Formula for slope is derived from the expression minSUM(y observed value - y Predicted value)^2 using d

667.

Calculate Slope and Y-Intercept for Regression Line long hand.

Calculate Slope using the SLOPE Function

Calculate the y-Intercept using the INTERCEPT Function

Slope = Rise Over Run = For every one unit of x, how far does y move?

Y-intercept = y value where x = zero. = point at which line crosses axis

Use slope and y-intercept to create estimated simple linear regression equation (lin

From sample data, the slope and y-intercept are point estimates for the population parameters f

Use estimated simple linear regression line to make predictions

Be careful when making predictions with the estimated simple linear regression equation (line or model)

range of the sample data. Why? Because the data may show a linear relationship over the range of sampl

relationship outside that sampled range.See how to use FORECAST function to make predictions.

Xbar

59.93333333

Example 2: 59.93333333

x

0

Mean: 59.93333333 $5,409 100

Count 15

n -1 14

Sample Point x y (x Deviation) (y Deviation) (x Deviation)^2

No.

Temperature

(F) Sales Chicken Soup (x - Xbar) (y - Ybar) (x - Xbar)^2

1 86 $3,300 26.06666667 -2109.33333 679.4711111

2 40 $8,200 -19.93333333 2790.666667 397.3377778

3 41 $8,900 -18.93333333 3490.666667 358.4711111

4 78 $3,100 18.06666667 -2309.33333 326.4044444

5 71 $4,020 11.06666667 -1389.33333 122.4711111

6 91 $1,950 31.06666667 -3459.33333 965.1377778

7 70 $2,500 10.06666667 -2909.33333 101.33777788 37 $6,500 -22.93333333 1090.666667 525.9377778

9 65 $6,210 5.066666667 800.6666667 25.67111111

10 42 $5,250 -17.93333333 -159.333333 321.6044444

11 53 $7,200 -6.933333333 1790.666667 48.07111111

12 83 $2,750 23.06666667 -2659.33333 532.0711111

13 63 $7,150 3.066666667 1740.666667 9.404444444

14 36 $7,900 -23.93333333 2490.666667 572.8044444

15 43 $6,210 -16.93333333 800.6666667 286.7377778

$0

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

0 20 40

Sa

lesC

hic

kenSoup

Tem


29/61


SUM Deviations^2 ====================>> 5272.933333

SUM Mult. Deviations ===========================================

Sample SD x 19.40716608 19.40716608

Sample SD y 2301.013648 2301.013648

Sample Covariance -37874.3333 -37874.33333

Coefficient of Correlation -0.84813245 -0.84813245Slope

Y-Intercept

x-value to make

prediction 71

Equation to Predict

Xbar y

66.27272727 0

Example 3: 66.27272727 8000

Sample Point x y

No.

Temperature

(F) Sales Ice Cream

1 91 $7,113

2 45 $2,044

3 46 $1,108

4 83 $7,093

5 76 $3,9026 96 $6,676

7 75 $5,403

8 42 $886

9 70 $4,740

10 47 $2,637

11 58 $3,150

Mean: 66.27272727 $4,068

Sample

Covariance

Coefficient of

Correlation Strength and Direction of the relationship (-1 to 0 to +1)

r^2 Coefficient of Determination = R^2 = "Goodness of fit for our line" (Number

Slope for every one unit of x, how far does y move?

Y Intercept Point at which estimated regression line crosses y-axis

x 85

Predicted y

y = 112x - 3354.1

R = 0.9056

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

0 20 40 60

Sa

lesIceCream

Temp



30/61

Check: $6,165.78


31/61

o Make Predictions

ifferential calculus. See text page

or model)

or slope and y-intercept

hen the x values are outside the

data, but may show some other

y

0

10000

Ybar

$5,409

$5,409

(y Deviation)^2

(x Deviation)*

(y Deviation)

(y - Ybar)^2

(x Deviation)*

(y Deviation)

4449287.111 -54983.28889

7787820.444 -55627.28889

12184753.78 -66089.95556

5333020.444 -41721.95556

1930247.111 -15375.28889

11966987.11 -107469.9556

8464220.444 -29287.288891189553.778 -25012.62222

641067.1111 4056.711111

25387.11111 2857.377778

3206487.111 -12415.28889

7072053.778 -61341.95556

3029920.444 5338.044444

6203420.444 -59609.95556

641067.1111 -13557.95556

y = -100.56x + 11436

R = 0.7193

60 80 100 120

perature (F)

Coefficient of Correlation = Measures Strength and Direction

Not Have A Problem With Units. Range From -1 to 0 to + 1. -1

Relationship (as x increases, y decreases). 0 = No Relationship.

Relationship (as x increases, y increases). Used for Linear Rela

rxy =

( )

sxsy

Sample Standard Deviation = Spread In Data. How Fairly Do

Data Points?

s =

2

(1)

Sample Covariance = Measure the Strength of the Linear Relati

but has problem with units. Note: See 4 Quadrant Example of

sense.

sxy =

(

)

1

Estimated Simple Linear Regression Equation

i = b0 + b1xi

Model based off of proof that minimizes:

Least Squares Criterion:

min= ( i)2

or min= ( b0 + b1xi)2

In order to get formula for b0 and b1:

Slope of Line (for every 1 unit of x, how much does y move?)

b1 =

(

)

2


32/61

74125293.33

=>> -530240.6667

Correlation is not causation Strength and Direction of the relationship (-1 toFor every one unit of x, how far does y move?

Point at which estimated regression line crosses y-axis

x Ybar

0 $4,068

120 $4,068

between 0 and 1)

80 100 120 140

erature (F)


Y-Intercept (at what point does the line cross the y-axis?)

b0 = Ybar - b1*Xbar


33/61


34/61

f Liner Relationship. Does

= Perfect Indirect (Negative)

+1 = Perfect Direct (Positive)

tionship only.

es The Mean Represent The

onship Between 2 Variables,

hy this measure makes


35/61

to +1)


36/61

Linear Regression #4: Calculate Slope & Y-Intercept, Create Estimated Equation and Use I

Formula for slope is derived from the expression minSUM(y observed value - y Predicted value)^2 usin

667.

Calculate Slope and Y-Intercept for Regression Line long hand.

Calculate Slope using the SLOPE Function

Calculate the y-Intercept using the INTERCEPT Function

Slope = Rise Over Run = For every one unit of x, how far does y move

Y-intercept = y value where x = zero. = point at which line crosses axi

Use slope and y-intercept to create estimated simple linear regression equation (li

From sample data, the slope and y-intercept are point estimates for the population parameter

Use estimated simple linear regression line to make predictions

Be careful when making predictions with the estimated simple linear regression equation (line or model

range of the sample data. Why? Because the data may show a linear relationship over the range of s

other relationship outside that sampled range.See how to use FORECAST function to make predictions.

Xbar

59.93333333

Example 2: 59.93333333

x

0

Mean: 59.93333 $5,409 100

Count 15

n -1 14

Sample Point x y (x Deviation) (y Deviation) (x Deviation)^2

No.

Temperat

ure (F) Sales Chicken Soup (x - Xbar) (y - Ybar) (x - Xbar)^2

1 86 $3,300 26.06666667 -2109.33333 679.4711111

2 40 $8,200 -19.93333333 2790.666667 397.3377778

3 41 $8,900 -18.93333333 3490.666667 358.4711111

4 78 $3,100 18.06666667 -2309.33333 326.4044444

5 71 $4,020 11.06666667 -1389.33333 122.4711111

6 91 $1,950 31.06666667 -3459.33333 965.1377778

7 70 $2,500 10.06666667 -2909.33333 101.3377778

8 37 $6,500 -22.93333333 1090.666667 525.9377778

9 65 $6,210 5.066666667 800.6666667 25.6711111110 42 $5,250 -17.93333333 -159.333333 321.6044444

11 53 $7,200 -6.933333333 1790.666667 48.07111111

12 83 $2,750 23.06666667 -2659.33333 532.0711111

13 63 $7,150 3.066666667 1740.666667 9.404444444

14 36 $7,900 -23.93333333 2490.666667 572.8044444

15 43 $6,210 -16.93333333 800.6666667 286.7377778


SUM Deviations^2 ====================>> 5272.933333

$0

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

0 20 40

Sa

lesC

hic

kenSoup

Tem


37/61

SUM Mult. Deviations ===========================================

Sample SD x 19.40716608 19.40716608

Sample SD y 2301.013648 2301.013648

Sample Covariance -37874.3333 -37874.33333

Coefficient of Correlation -0.84813245 -0.84813245

Slope -100.558955 -100.5589552

Y-Intercept $11,436.17 11436.16671x-value to make

prediction 71 $4,296.48 4296.480896

Equation to Predict y Predicted = $11436.17 - $100.56*x

y Predicted = 11436.17 + -100.56*x

Xbar y

66.27272727 0

Example 3: 66.27272727 8000

Sample Point x y

No.

Temperat


1 91 $7,113

2 45 $2,044

3 46 $1,108

4 83 $7,093

5 76 $3,902

6 96 $6,676

7 75 $5,4038 42 $886

9 70 $4,740

10 47 $2,637

11 58 $3,150

Mean: 66.27273 $4,068

Sample

Covariance 43143.69

Coefficient of

Correlation 0.951608 Strength and Direction of the relationship (-1 to 0 to +1)

r^2 0.905558 Coefficient of Determination = R^2 = "Goodness of fit for our line" (Number

Slope 111.9981 for every one unit of x, how far does y move?

Y Intercept -3354.05 Point at which estimated regression line crosses y-axis

x 85

Predicted y 6165.782

Check: 6165.782

y = 112x - 3354.1

R = 0.9056

$0

$1,000

$2,000$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

0 20 40 60

Sa

lesIceCream

Temp



38/61

t To Make Predictions

differential calculus. See text page

?

ine or model)

s for slope and y-intercept

) when the x values are outside the

mple data, but may show some

y

0

10000

Ybar

$5,409

$5,409

(y Deviation)^2

(x Deviation)*

(y Deviation)

(y - Ybar)^2

(x Deviation)*

(y Deviation)

4449287.111 -54983.28889

7787820.444 -55627.28889

12184753.78 -66089.95556

5333020.444 -41721.95556

1930247.111 -15375.28889

11966987.11 -107469.9556

8464220.444 -29287.28889

1189553.778 -25012.62222

641067.1111 4056.71111125387.11111 2857.377778

3206487.111 -12415.28889

7072053.778 -61341.95556

3029920.444 5338.044444

6203420.444 -59609.95556

641067.1111 -13557.95556

74125293.33

y = -100.56x + 11436

R = 0.7193

60 80 100 120

perature (F)

Coefficient of Correlation = Measures Strength and Direction

Not Have A Problem With Units. Range From -1 to 0 to + 1. -1Relationship (as x increases, y decreases). 0 = No Relationship.

Relationship (as x increases, y increases). Used for Linear Rela

rxy =

( )

sxsy

Sample Standard Deviation = Spread In Data. How Fairly Do

Data Points?

s =

2

(1)

Sample Covariance = Measure the Strength of the Linear Relati

but has problem with units. Note: See 4 Quadrant Example of

sense.

sxy =

(

)

1

Estimated Simple Linear Regression Equation

i = b0 + b1xiModel based off of proof that minimizes:

Least Squares Criterion:

min= ( i)2

or min= ( b0 + b1xi)2

In order to get formula for b0 and b1:

Slope of Line (for every 1 unit of x, how much does y move?)

b1 =

(

)

2

Y-Intercept (at what point does the line cross the y-axis?)


39/61

=>> -530240.6667

Correlation is not causation Strength and Direction of the relationship (-1 to

For every one unit of x, how far does y move?

Point at which estimated regression line crosses y-axis

x Ybar

0 $4,068

120 $4,068

between 0 and 1)

80 100 120 140

erature (F)


0 = bar - 1 bar


40/61

f Liner Relationship. Does

= Perfect Indirect (Negative)+1 = Perfect Direct (Positive)

tionship only.

es The Mean Represent The

onship Between 2 Variables,

hy this measure makes


41/61

to +1)


42/61

Linear Regression #5: Coefficient of Determination: Goodness of Fit =

Calculate Total Sum Of Squares (Total Y Deviations Squared) = SST = How well observations cluster around

deviations of y observed and Mean of Y (Ybar)

Calculate Sum of Squares Due To Error = SSE = How well observations cluster around estimated simple line

deviations between y observed and y predicted = measure of variation that is not explained by the estimat

model).Calculate Sum of Squares Due To Regression = SSR = SST - SSE = sum of squares of deviations betwe

Relationship between SST and SSR and SST is: SST = SSR + SSE. When there is no error, the predicted values

regression line and therefore SSE would equal zero. In this case SST = SSR + 0 and SSR/SST = 1, which mean

the Coefficient of Determination will always be a number between 0 and 1. 0 = "no goodness of fit

SSR/SST = Coefficient of Determination = R Squared = r^2

Use RSQ function to calculate Coefficient of Determination

Use Coefficient of Correlation Squared to calculate coefficient of Deter

Coefficient of Determination can be used for linear and non-linear relationships. This is compared to Coeffic

for linear relationships.

Xbar Ybar

Mean 59.93333 $5,409

Slope -100.559 Part of Total Variation

Intercept 11436.17 Not explained by model

Sample Point x y Predicted y Residual Residual^2

No.

Temperat

ure (F)

Sales Chicken

Soup Predicted y

(y Observed - y

Predicted)

(y Observed - y

Predicted)^2

1 86 $3,300

2 40 $8,200

3 41 $8,900

4 78 $3,100

5 71 $4,020

6 91 $1,950

7 70 $2,500

8 37 $6,500

9 65 $6,210

10 42 $5,25011 53 $7,200

12 83 $2,750

13 63 $7,150

14 36 $7,900

15 43 $6,210

SSE

SSR

$0

$2,000

$4,000

$6,000

$8,000

$10,000

0 20 40 60

Sa

lesC

hic

ke

nSoup

Temperature

Sales Chicken Soup

Xbar

Yabr

Observation 3 Total Variation (y3

Residual (y3 - Y Observed)

Explained Part of Total Variation (

Linear (Sales Chicken Soup)


43/61

SSR + SSE = SST

Coefficient of Determination = r^2 = Measure of goodness of fit = r^2 = SSR/SST

Check:


r^2 = SSR/SST

Proportion of the variability in the dependent variable y that is explained by the estimated rHow well does the estimated regression line fit the data?

Measure of the goodness of fit for the estimated regression line

A number between 0 and +1

Can be used your nonlinear relationships as well as linear.

How well are observations are more closely grouped about the least squares line? 1 = perfec

Xbar y

66.27272727 0

Example 3: 66.27272727 8000

Sample Point x y

No.

Temperat


1 91 $7,113

2 45 $2,044

3 46 $1,108

4 83 $7,093

5 76 $3,902

6 96 $6,676

7 75 $5,403

8 42 $8869 70 $4,740

10 47 $2,637

11 58 $3,150

Mean: 66.27273 4068.363636


y = 112x - 3354.1

R = 0.9056

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

0 20 40 60

Sa

lesIceCream

Temp



44/61


x 75


Coefficient of


r^2 Coefficient of Determination = R^2 = "Goodness of fit for our line" (Number betwProportion of the variability in the dependent variable y that is explained by the e

How well does the estimated regression line fit the data?



45/61

SR/SST Xbar

Bar (Y Mean Plotted Line) = Total squared59.93333

ar regression equation = sum of squares of

d simple linear regression equation (line or59.93333

n y predicted and Mean of Y (Ybar)

nd the observed values would all lie on the

perfect "goodness of fit". This means that

and 1 = "perfect goodness of fit".

ination

ient of Correlation, which can only be used

Part of Total Variation

Explained by Model

(Predicted y - Ybar)^2 (y Deviations)^2

(Predicted y - Ybar)^2

(y Observed -

Ybar)^2

SST = Total Variation

ith residual = observed value - predicte

predict = ( i)

Sum Of Squares Due To Error (in model

the Estimated Line = SSE = Not Explain

SSE = ( i)2

Total Sum Of Squares (Deviation from

cluster around the Ybar Line = SST

SST = ( )2

Sum Of Squares Due To Regression (Pr

SST = SSR

SSR = (i )2

Relationship between three:

SST = SSR + SSE

If there is no deviation in the observed

SSR = SST, thus:

SSR/SST = 1 = perfect PrCoefficient of Determination = How we

the data? = Measure of the goodness o

number between 0 and +1. Can be use

linear.

y = -100.56x + 11436

R = 0.7193

80 100 120

(F)

- Ybar)

Y Predicted - Ybar)


46/61

Goodness of fit of model to observed values (number between

Strength and Direction (number between -1 and 1)

gression equation

tly. 0 = Not at all.

x Ybar

0 $4,068

120 $4,068

rxy2 = r2 = SSR/SST

= The percentage of total sum of squar

estimated regression equation = Propo

variable y that is explained by the estiare more closely grouped about the lea

Using r^2 only, we can draw no conclu

between x and y is statistically significa

considerations that involve sample size

sampling distributions of the least squa

80 100 120 140

rature (F)

bar Linear (Sales Ice Cream)


47/61

een 0 and 1)stimated regression equation


48/61

y x Yabr

Observation 3 Total

Variation (y3 - Ybar)

Residual (y3 - Y

Observed) y

1 0 $5,409 41 $8,900 42 $8,900

10000 100 $5,409 41 $5,409 42 7313.249551

d value = represents error in using i to

) = How well observations cluster around

d Part of SST

ean) = How well the observations

dicted y minus Y bar) = Explained Part of

values and the model values SSE = 0 and

diction.ll does the estimated regression line fit

f fit for the estimated regression line = A

your nonlinear relationships as well as


49/61

that can be explained by using the

rtion of the variability in the dependent

ated regression equation. Observationsst squares line.

ion about whether the relationship

nt. Such conclusions must be based on

and properties of the appropriate

res estimators.


50/61


51/61

Explained Part of Total

Variation (Y Predicted - Ybar) y

42 7313.25

42 $5,409


52/61

Linear Regression #5: Coefficient of Determination: Goodness of Fit =

Calculate Total Sum Of Squares (Total Y Deviations Squared) = SST = How well observations cluster around

deviations of y observed and Mean of Y (Ybar)

Calculate Sum of Squares Due To Error = SSE = How well observations cluster around estimated simple line

deviations between y observed and y predicted = measure of variation that is not explained by the estimat

model).Calculate Sum of Squares Due To Regression = SSR = SST - SSE = sum of squares of deviations betwe

Relationship between SST and SSR and SST is: SST = SSR + SSE. When there is no error, the predicted values

regression line and therefore SSE would equal zero. In this case SST = SSR + 0 and SSR/SST = 1, which mean

the Coefficient of Determination will always be a number between 0 and 1. 0 = "no goodness of fitSSR/SST = Coefficient of Determination = R Squared = r^2

Use RSQ function to calculate Coefficient of Determination

Use Coefficient of Correlation Squared to calculate coefficient of Deter

Coefficient of Determination can be used for linear and non-linear relationships. This is compared to Coeffic

for linear relationships.

Xbar Ybar

Mean 59.93333 $5,409

Slope -100.559 Part of Total Variation

Intercept 11436.17 Not explained by model

Sample Point x y Predicted y Residual Residual^2

No.

Temperat

ure (F)

Sales Chicken

Soup Predicted y

(y Observed - y

Predicted)

(y Observed - y

Predicted)^2

1 86 $3,300 2788.096569 511.9034314 262045.1232 40 $8,200 7413.808506 786.1914937 618097.0647

3 41 $8,900 7313.249551 1586.750449 2517776.987

4 78 $3,100 3592.56821 -492.56821 242623.4415

5 71 $4,020 4296.480896 -276.4808961 76441.68594

6 91 $1,950 2285.301793 -335.3017928 112427.2923

7 70 $2,500 4397.039851 -1897.039851 3598760.197

8 37 $6,500 7715.485372 -1215.485372 1477404.689

9 65 $6,210 4899.834627 1310.165373 1716533.304

10 42 $5,250 7212.690596 -1962.690596 3852154.376

11 53 $7,200 6106.542089 1093.457911 1195650.203

12 83 $2,750 3089.773434 -339.7734341 115445.986513 63 $7,150 5100.952537 2049.047463 4198595.504

14 36 $7,900 7816.044327 83.955673 7048.555028

15 43 $6,210 7112.131641 -902.1316408 813841.4974

SSE 20804845.91

SSR

SSR + SSE = SST

$0

$2,000

$4,000

$6,000

$8,000

$10,000

0 20 40 60

Sa

lesCh

ickenSoup

Temperature

Sales Chicken SoupXbarYabrObservation 3 Total Variation (y3Residual (y3 - Y Observed)Explained Part of Total Variation (Linear (Sales Chicken Soup)


53/61

Coefficient of Determination = r^2 = Measure of goodness of fit = r^2 = SSR/SST

Check:


r^2 = SSR/SST

Proportion of the variability in the dependent variable y that is explained by the estimated r

How well does the estimated regression line fit the data?

Measure of the goodness of fit for the estimated regression lineA number between 0 and +1


How well are observations are more closely grouped about the least squares line? 1 = perfec

Xbar y

66.27272727 0

Example 3: 66.27272727 8000

Sample Point x y


1 91 $7,113

2 45 $2,044

3 46 $1,108

4 83 $7,093

5 76 $3,902

6 96 $6,676

7 75 $5,403

8 42 $886

9 70 $4,740

10 47 $2,63711 58 $3,150

Mean: 66.27273 4068.363636



x 75

y = 112x - 3354.1R = 0.9056

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

0 20 40 60

Sa

lesIceCream

Temp



54/61


Coefficient of


r^2 0.905558 Coefficient of Determination = R^2 = "Goodness of fit for our line" (Number betw

Proportion of the variability in the dependent variable y that is explained by the e

How well does the estimated regression line fit the data?Can be used your nonlinear relationships as well as linear.


55/61

SR/SST Xbar

Bar (Y Mean Plotted Line) = Total squared59.93333

ar regression equation = sum of squares of

d simple linear regression equation (line or59.93333

n y predicted and Mean of Y (Ybar)

nd the observed values would all lie on the

perfect "goodness of fit". This means that

and 1 = "perfect goodness of fit".

ination

ient of Correlation, which can only be used

Part of Total Variation

Explained by Model

(Predicted y - Ybar)^2 (y Deviations)^2

(Predicted y - Ybar)^2

(y Observed -

Ybar)^2

6870882.177 4449287.1114017920.719 7787820.444

3624896.965 12184753.78

3300635.513 5333020.444

1238440.547 1930247.111

9759573.066 11966987.11

1024738.094 8464220.444

5318337.225 1189553.778

259588.9316 641067.1111

3252097.417 25387.11111

486100.0492 3206487.111

5380358.126 7072053.77895098.71525 3029920.444

5792257.807 6203420.444

2899522.076 641067.1111

74125293.33 SST = Total Variation

53320447.43

74125293.33

ith residual = observed value - predicte

predict = ( i)

Sum Of Squares Due To Error (in model

the Estimated Line = SSE = Not Explain

SSE = ( i)2

Total Sum Of Squares (Deviation from

cluster around the Ybar Line = SST

SST = ( )2

Sum Of Squares Due To Regression (Pr

SST = SSR

SSR = (i )2

Relationship between three:

SST = SSR + SSE

If there is no deviation in the observed

SSR = SST, thus:

SSR/SST = 1 = perfect PrCoefficient of Determination = How we

the data? = Measure of the goodness o

number between 0 and +1. Can be use

linear.

y = -100.56x + 11436

R = 0.7193

80 100 120

(F)

- Ybar)

Y Predicted - Ybar)


56/61

0.719328653 Goodness of fit of model to observed values (number between

0.719328653

-0.84813245 Strength and Direction (number between -1 and 1)

0.719328653

gression equation

tly. 0 = Not at all.

x Ybar

0 $4,068

120 $4,068

rxy2 = r2 = SSR/SST

= The percentage of total sum of squar

estimated regression equation = Propo

variable y that is explained by the estiare more closely grouped about the lea

Using r^2 only, we can draw no conclu

between x and y is statistically significa

considerations that involve sample size

sampling distributions of the least squa

80 100 120 140

rature (F)

bar Linear (Sales Ice Cream)


57/61

een 0 and 1)

stimated regression equation


58/61

y x Yabr

Observation 3 Total

Variation (y3 - Ybar)

Residual (y3 - Y

Observed) y

1 0 $5,409 41 $8,900 42 $8,900

10000 100 $5,409 41 $5,409 42 7313.249551

d value = represents error in using i to

) = How well observations cluster around

d Part of SST

ean) = How well the observations

dicted y minus Y bar) = Explained Part of

values and the model values SSE = 0 and

diction.ll does the estimated regression line fit

f fit for the estimated regression line = A

your nonlinear relationships as well as


59/61

that can be explained by using the

rtion of the variability in the dependent

ated regression equation. Observationsst squares line.

ion about whether the relationship

nt. Such conclusions must be based on

and properties of the appropriate

res estimators.


60/61


61/61

Explained Part of Total

Variation (Y Predicted - Ybar) y

42 7313.25

42 $5,409

Documents

Busn210ch14 Statistics Series