+ All Categories
Home > Documents > SLIDES . BY

SLIDES . BY

Date post: 03-Jan-2016
Category:
Upload: william-strickland
View: 27 times
Download: 0 times
Share this document with a friend
Description:
SLIDES . BY. John Loucks St . Edward’s University. Chapter 12, Part A Simple Linear Regression. Simple Linear Regression Model. Least Squares Method. Coefficient of Determination. Model Assumptions. Testing for Significance. Simple Linear Regression. - PowerPoint PPT Presentation
Popular Tags:
39
1 Slide Cengage Learning. All Rights Reserved. May not be scanned, copied duplicated, or posted to a publicly accessible website, in whole or in part. SLIDES . BY John Loucks St. Edward’s University . . . . . . . . . . .
Transcript
Page 1: SLIDES . BY

1 1 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

SLIDES . BY

John LoucksSt. Edward’sUniversity

...........

Page 2: SLIDES . BY

2 2 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 12, Part ASimple Linear Regression

Simple Linear Regression Model Least Squares Method Coefficient of Determination Model Assumptions Testing for Significance

Page 3: SLIDES . BY

3 3 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Simple Linear Regression

Regression analysis can be used to develop an equation showing how the variables are related.

Managerial decisions often are based on the relationship between two or more variables.

The variables being used to predict the value of the dependent variable are called the independent variables and are denoted by x.

The variable being predicted is called the dependent variable and is denoted by y.

Page 4: SLIDES . BY

4 4 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Simple Linear Regression

The relationship between the two variables is approximated by a straight line.

Simple linear regression involves one independent variable and one dependent variable.

Regression analysis involving two or more independent variables is called multiple regression.

Page 5: SLIDES . BY

5 5 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Simple Linear Regression Model

y = b0 + b1x +e

where: b0 and b1 are called parameters of the model,

e is a random variable called the error term.

The simple linear regression model is:

The equation that describes how y is related to x and an error term is called the regression model.

Page 6: SLIDES . BY

6 6 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Simple Linear Regression Equation

The simple linear regression equation is:

• E(y) is the expected value of y for a given x value.

• b1 is the slope of the regression line.

• b0 is the y intercept of the regression line.

• Graph of the regression equation is a straight line.

E(y) = 0 + 1x

Page 7: SLIDES . BY

7 7 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Simple Linear Regression Equation

Positive Linear Relationship

E(y)

x

Slope b1

is positive

Regression line

Intercept b0

Page 8: SLIDES . BY

8 8 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Simple Linear Regression Equation

Negative Linear Relationship

E(y)

x

Slope b1

is negative

Regression lineIntercept

b0

Page 9: SLIDES . BY

9 9 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Simple Linear Regression Equation

No Relationship

E(y)

x

Slope b1

is 0

Regression lineIntercept b0

Page 10: SLIDES . BY

10 10 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Estimated Simple Linear Regression Equation

The estimated simple linear regression equation

0 1y b b x

• is the estimated value of y for a given x value.y• b1 is the slope of the line.

• b0 is the y intercept of the line.

• The graph is called the estimated regression line.

Page 11: SLIDES . BY

11 11 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Estimation Process

Regression Modely = b0 + b1x +e

Regression EquationE(y) = b0 + b1x

Unknown Parametersb0, b1

Sample Data:x y

x1 y1

. . . . xn yn

b0 and b1

provide estimates ofb0 and b1

EstimatedRegression Equation

Sample Statistics

b0, b1

0 1y b b x

Page 12: SLIDES . BY

12 12 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Least Squares Method

Least Squares Criterion

min (y yi i )2

where:yi = observed value of the dependent variable

for the ith observation^yi = estimated value of the dependent variable

for the ith observation

Page 13: SLIDES . BY

13 13 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Slope for the Estimated Regression Equation

1 2

( )( )

( )i i

i

x x y yb

x x

Least Squares Method

where:xi = value of independent variable for ith observation

_y = mean value for dependent variable

_x = mean value for independent variable

yi = value of dependent variable for ith observation

Page 14: SLIDES . BY

14 14 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

y-Intercept for the Estimated Regression Equation

Least Squares Method

0 1b y b x

Page 15: SLIDES . BY

15 15 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Reed Auto periodically has a special week-long sale.

As part of the advertising campaign Reed runs one or

more television commercials during the weekendpreceding the sale. Data from a sample of 5

previoussales are shown on the next slide.

Simple Linear Regression

Example: Reed Auto Sales

Page 16: SLIDES . BY

16 16 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Simple Linear Regression

Example: Reed Auto Sales

Number of TV Ads (x)

Number ofCars Sold (y)

13213

1424181727

Sx = 10 Sy = 1002x 20y

Page 17: SLIDES . BY

17 17 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Estimated Regression Equation

ˆ 10 5y x

1 2

( )( ) 205

( ) 4i i

i

x x y yb

x x

0 1 20 5(2) 10b y b x

Slope for the Estimated Regression Equation

y-Intercept for the Estimated Regression Equation

Estimated Regression Equation

Page 18: SLIDES . BY

18 18 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Coefficient of Determination

Relationship Among SST, SSR, SSE

where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error

SST = SSR + SSE

2( )iy y 2ˆ( )iy y 2ˆ( )i iy y

Page 19: SLIDES . BY

19 19 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

The coefficient of determination is:

Coefficient of Determination

where:SSR = sum of squares due to regressionSST = total sum of squares

r2 = SSR/SST

Page 20: SLIDES . BY

20 20 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Coefficient of Determination

r2 = SSR/SST = 100/114 = .8772

The regression relationship is very strong; 87.72%of the variability in the number of cars sold can beexplained by the linear relationship between thenumber of TV ads and the number of cars sold.

Page 21: SLIDES . BY

21 21 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Sample Correlation Coefficient

21 ) of(sign rbrxy

ionDeterminat oft Coefficien ) of(sign 1brxy

where: b1 = the slope of the estimated regression

equation xbby 10ˆ

Page 22: SLIDES . BY

22 22 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

21 ) of(sign rbrxy

The sign of b1 in the equation is “+”.ˆ 10 5y x

=+ .8772xyr

Sample Correlation Coefficient

rxy = +.9366

Page 23: SLIDES . BY

23 23 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Assumptions About the Error Term e

1. The error is a random variable with mean of zero.

2. The variance of , denoted by 2, is the same for all values of the independent variable.

3. The values of are independent.

4. The error is a normally distributed random variable.

Page 24: SLIDES . BY

24 24 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Testing for Significance

To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of b1 is zero.

Two tests are commonly used:

t Test and F Test

Both the t test and F test require an estimate of s 2, the variance of e in the regression model.

Page 25: SLIDES . BY

25 25 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

An Estimate of s 2

Testing for Significance

210

2 )()ˆ(SSE iiii xbbyyy

where:

s 2 = MSE = SSE/(n - 2)

The mean square error (MSE) provides the estimateof s 2, and the notation s2 is also used.

Page 26: SLIDES . BY

26 26 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Testing for Significance

An Estimate of s

2

SSEMSE

n

s

• To estimate s we take the square root of s 2.

• The resulting s is called the standard error of the estimate.

Page 27: SLIDES . BY

27 27 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Hypotheses

Test Statistic

Testing for Significance: t Test

0 1: 0H

1: 0aH

1

1

b

bt

s where

1 2( )b

i

ss

x x

Page 28: SLIDES . BY

28 28 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Rejection Rule

Testing for Significance: t Test

where: t is based on a t distribution

with n - 2 degrees of freedom

Reject H0 if p-value < a or t < -tor t > t

Page 29: SLIDES . BY

29 29 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

1. Determine the hypotheses.

2. Specify the level of significance.

3. Select the test statistic.

a = .05

4. State the rejection rule.Reject H0 if p-value < .05or |t| > 3.182 (with

3 degrees of freedom)

Testing for Significance: t Test

0 1: 0H

1: 0aH

1

1

b

bt

s

Page 30: SLIDES . BY

30 30 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Testing for Significance: t Test

5. Compute the value of the test statistic.

6. Determine whether to reject H0.

t = 4.541 provides an area of .01 in the upper tail. Hence, the p-value is .02.

(Also, t = 4.63 > 3.182.) We can reject H0.

1

1 54.63

1.08b

bt

s

Page 31: SLIDES . BY

31 31 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Confidence Interval for 1

H0 is rejected if the hypothesized value of 1 is not included in the confidence interval for 1.

We can use a 95% confidence interval for 1 to test the hypotheses just used in the t test.

Page 32: SLIDES . BY

32 32 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

The form of a confidence interval for 1 is:

Confidence Interval for 1

11 / 2 bb t s

where is the t value providing an areaof a/2 in the upper tail of a t distributionwith n - 2 degrees of freedom

2/tb1 is the

pointestimat

or

is themarginof error

1/ 2 bt s

Page 33: SLIDES . BY

33 33 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Confidence Interval for 1

Reject H0 if 0 is not included in

the confidence interval for 1.

0 is not included in the confidence interval. Reject H0

= 5 +/- 3.182(1.08) = 5 +/- 3.4412/1 bstb

or 1.56 to 8.44

Rejection Rule

95% Confidence Interval for 1

Conclusion

Page 34: SLIDES . BY

34 34 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Hypotheses

Test Statistic

Testing for Significance: F Test

F = MSR/MSE

0 1: 0H

1: 0aH

Page 35: SLIDES . BY

35 35 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Rejection Rule

Testing for Significance: F Test

where:F is based on an F distribution with

1 degree of freedom in the numerator andn - 2 degrees of freedom in the denominator

Reject H0 if p-value < a

or F > F

Page 36: SLIDES . BY

36 36 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

1. Determine the hypotheses.

2. Specify the level of significance.

3. Select the test statistic.

a = .05

4. State the rejection rule.Reject H0 if p-value < .05or F > 10.13 (with 1 d.f.

in numerator and 3 d.f. in denominator)

Testing for Significance: F Test

0 1: 0H

1: 0aH

F = MSR/MSE

Page 37: SLIDES . BY

37 37 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Testing for Significance: F Test

5. Compute the value of the test statistic.

6. Determine whether to reject H0.

F = 17.44 provides an area of .025 in the upper tail. Thus, the p-value corresponding to F = 21.43is less than .025. Hence, we reject H0.

F = MSR/MSE = 100/4.667 = 21.43

The statistical evidence is sufficient to concludethat we have a significant relationship between thenumber of TV ads aired and the number of cars sold.

Page 38: SLIDES . BY

38 38 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

Some Cautions about theInterpretation of Significance Tests

Just because we are able to reject H0: b1 = 0 and demonstrate statistical significance does not enable

us to conclude that there is a linear relationshipbetween x and y.

Rejecting H0: b1 = 0 and concluding that the

relationship between x and y is significant does not enable us to conclude that a cause-and-effect

relationship is present between x and y.

Page 39: SLIDES . BY

39 39 Slide Slide© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied

or duplicated, or posted to a publicly accessible website, in whole or in part.

End of Chapter 12, Part A


Recommended