Home > Chapter 12 - Simple Linear Regressions

Chapter 12 - Simple Linear Regressions


 
 

1  

Chapter 12 
Simple Linear Regression 

  • Simple Linear Regression Model
  • Least Squares Method 
  • Coefficient of Determination
  • Model Assumptions
  • Testing for Significance
  • Using the Estimated Regression Equation

       for Estimation and Prediction

  • Computer Solution
  • Residual Analysis: Validating Model Assumptions
 
 

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

y = b0 + b1x +e 
 

    • b0 and b1 are called parameters of the model.
    • e  is a random variable called the error term.
 

Simple Linear Regression Model


 
 

3  

  • The simple linear regression equation is:
 

E(y) = 0 + 1x 
 

    • Graph of the regression equation is a straight line.
    • b0 is the y intercept of the regression line.
    • b1 is the slope of the regression line.
    • E(y) is the expected value of y for a given x value.
 

Simple Linear Regression Equation


 
 

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Simple Linear Regression Equation 

  • Positive Linear Relationship
 

E(y

x 

Slope b1

is positive 

Regression line 

Intercept

             b0


 
 

5  

Simple Linear Regression Equation 

  • Negative Linear Relationship
 

E(y

x 

Slope b1

is negative 

Regression line 

Intercept

             b0


 
 

6  

Simple Linear Regression Equation 

  • No Relationship
 

E(y

x 

Slope b1

is 0 

Regression line 

Intercept

             b0


 
 

7  

  • The estimated simple linear regression equation is:
    • The graph is called the estimated regression line. 
       
       
    • b0 is the y intercept of the line.
    • b1 is the slope of the line.
    •     is the estimated value of y for a given x value.
 
 

Estimated Simple Linear Regression Equation


 
 

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Estimation Process 
 

Regression Model

y = b0 + b1x +e

Regression Equation

E(y) = b0 + b1x

Unknown Parameters

b0, b1 

Sample Data:

x        y 

x1      y1

.       .

.       .

xn     yn 

Estimated

Regression Equation

 

Sample Statistics

b0, b1 

b0 and b1

provide estimates of

b0 and b1


 
 

9  

  • Least Squares Criterion
 
 
 

    where:

          yi = observed value of the dependent variable

                 for the ith observation

          yi = estimated value of the dependent variable

                 for the ith observation 
 

Least Squares Method 

^


 
 

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  • Slope for the Estimated Regression Equation
 
 

The Least Squares Method


 
 

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  • y-Intercept for the Estimated Regression Equation
 
 
 

    where:

    xi = value of independent variable for ith observation

    yi = value of dependent variable for ith observation

   x = mean value for independent variable

      y = mean value for dependent variable

      n = total number of observations 



The Least Squares Method


 
 

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Example:  Reed Auto Sales 

  • Simple Linear Regression

                 Reed Auto periodically has a special week-long sale.  As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale.  Data from a sample of 5 previous sales are shown on the next slide.


 
 

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Example:  Reed Auto Sales 

  • Simple Linear Regression
 

          Number of TV Ads     Number of Cars Sold

          1 14

          3 24

          2 18

          1 17

          3 27


 
 

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  • Slope for the Estimated Regression Equation
 

                 b1  =  220 - (10)(100)/5  =  _____

                         24 - (10)2/5 

  • y-Intercept for the Estimated Regression Equation
 

                       b0  =  20 - 5(2)  =  _____ 

  • Estimated Regression Equation
 

y = 10 + 5x 


Example:  Reed Auto Sales


 
 

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Example:  Reed Auto Sales 

  • Scatter Diagram
 

^


 
 

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  • Relationship Among SST, SSR, SSE
 

SST = SSR + SSE 
 
 
 
 
 
 
 
 

    where:

               SST = total sum of squares

               SSR = sum of squares due to regression

               SSE = sum of squares due to error 

The Coefficient of Determination 


^


 
 

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  • The coefficient of determination is:
 

r2 = SSR/SST 

    where:

               SST = total sum of squares

               SSR = sum of squares due to regression 

The Coefficient of Determination


 
 

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  • Coefficient of Determination
 

                r2 = SSR/SST = 100/114 =

          The regression relationship is very strong because 88% of the variation in number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold. 

Example:  Reed Auto Sales


 
 

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The Correlation Coefficient 

  • Sample Correlation Coefficient
 
 
 
 

    where:

              b1 = the slope of the estimated regression

                equation


 
 

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  • Sample Correlation Coefficient
 
 

    The sign of b1 in the equation        is ��+��. 
 

                       rxy =    +.9366  
 
 

Example:  Reed Auto Sales


 
 

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Model Assumptions 

  • Assumptions About the Error Term
    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.
 
 

22  

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
    • F  Test
  • Both tests require an estimate of s 2, the variance of e in the regression model.
 
 

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  • An Estimate of s 2

    The mean square error (MSE) provides the estimate

    of s 2, and the notation s2 is also used.  

                         s2 = MSE = SSE/(n-2) 

    where: 

Testing for Significance


 
 

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Testing for Significance 

  • An Estimate of s
    • To estimate s  we take the square root of s 2.
    • The resulting s is called the standard error of the estimate.
 
 

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  • Hypotheses
 

                          H0: 1 = 0

                          Ha: 1 = 0 

  • Test Statistic
 
 
 

    where 
 

Testing for Significance:  t  Test


 
 

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  • Rejection Rule
 

Reject H0 if t < -tor t > t 
 

    where:   t is based on a t  distribution

            with n - 2 degrees of freedom 

Testing for Significance:  t  Test


 
 

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  • t  Test
    • Hypotheses

                                H0: 1 = 0

                          Ha: 1 = 0 

    • Rejection Rule

                    For = .05 and d.f. = 3,  t.025 = _____

                            Reject H0 if t > t.025 = _____ 

Example:  Reed Auto Sales


 
 

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  • t  Test
    • Test Statistics

                          t = _____/_____ = 4.63

    • Conclusions

              t = 4.63 > 3.182, so reject H0  

Example:  Reed Auto Sales


 
 

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Confidence Interval for

1 

  • We can use a 95% confidence interval for 1 to test the hypotheses just used in the t test.
  • H0 is rejected if the hypothesized value of  1 is not included in the confidence interval for  1.

 


 
 

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  • The form of a confidence interval for 1 is:
 
 

    

where      b1    is the point estimate

                      

is the margin of error

                      

is the t value providing an area

                      

of

a

/2 in the upper tail of a

       t  distribution with n - 2 degrees

                      

of freedom 
 

Confidence Interval for

1


 
 

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  • Rejection Rule

          

Reject H0 if 0 is not included in

          

the confidence interval for 

1.

  • 95% Confidence Interval for 1

                

        = 5 +/- 3.182(1.08) = 5 +/- 3.44 

                      

or    ____ to ____ 

  • Conclusion

    

0 is not included in the confidence interval. 

    Reject H0   

Example:  Reed Auto Sales


 
 

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  • Hypotheses
 

          

               H0:

1 = 0

          

              Ha:

1 = 0 

  • Test Statistic
 

F = MSR/MSE 

Testing for Significance: F  Test


 
 

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  • Rejection Rule
 

Reject H0 if F > F

 

    

where: F

is based on an F distribution

                

with 1 d.f. in the numerator and

                

n - 2 d.f. in the denominator 

Testing for Significance: F  Test


 
 

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  • F Test
    • Hypotheses

                                H0: 1 = 0

          

                Ha:

1 = 0

    • Rejection Rule

                    For = .05 and d.f. = 1, 3:  F.05 = ______

                           Reject H0 if F > F.05 = ______. 

Example:  Reed Auto Sales


 
 

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  • F Test
    • Test Statistic

    F = MSR/MSE = ____  / ______ = 21.43

    • Conclusion

                    F = 21.43 > 10.13, so we reject H0

Example:  Reed Auto Sales


 
 

36  

Some Cautions about the 
Interpretation of Significance Tests 

  • 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.
  • 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 relationship between x and y.
 
 

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  • Confidence Interval Estimate of E(yp)
  • Prediction Interval Estimate of yp 
     
 

yp + t

/2 sind 
 

    

where: confidence coefficient is 1 -

and

                

t

/2 is based on a t distribution

                

with n - 2 degrees of freedom 
 

Using the Estimated Regression Equation 
for Estimation and Prediction


 
 

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  • Point Estimation

    

If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: 

y = 10 + 5(3) = ______ cars 
 


Example:  Reed Auto Sales


 
 

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  • Confidence Interval for E(yp)

    

95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is: 

25 + 4.61  =  ______ to _______ cars 

Example:  Reed Auto Sales


 
 

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  • Prediction Interval for yp

    

95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: 

         25 + 8.28  =  _____ to ______ cars 

Example:  Reed Auto Sales


 
 

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  • Residual for Observation  i
 

yi

 yi 

  • Standardized Residual for Observation  i
 
 

    

where: 

    

and   

Residual Analysis 




^


 
 

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Example:  Reed Auto Sales 

  • Residuals
 
 

43  

Example:  Reed Auto Sales 

  • Residual Plot
 
 

44  

Residual Analysis 

  • Residual Plot
 

x 


Good Pattern 

Residual


 
 

45  

Residual Analysis 

  • Residual Plot
 

x 


Nonconstant Variance 

Residual


 
 

46  

Residual Analysis 

  • Residual Plot
 

x 


Model Form Not Adequate 

Residual


 
 

47  

End of Chapter 12


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