Chapter 1 Smple Lnear Regresson Smple Lnear Regresson Model Least Squares Method Coeffcent of Determnaton Model Assumptons Testng for Sgnfcance
Smple Lnear Regresson Model The equaton that descrbes how y s related to x and an error term s called the regresson model. The smple lnear regresson model s: y = β 0 + β 1 x +ε where: β 0 and β 1 are called parameters of the model, ε s a random varable called the error term.
Smple Lnear Regresson Equaton The smple lnear regresson equaton s: E(y) = β 0 + β 1 x Graph of the regresson equaton s a straght lne. β 0 s the y ntercept of the regresson lne. β 1 s the slope of the regresson lne. E(y) s the expected value of y for a gven x value.
Smple Lnear Regresson Equaton Postve Lnear Relatonshp E(y) Regresson lne Intercept β 0 Slope β 1 s postve x
Smple Lnear Regresson Equaton Negatve Lnear Relatonshp E(y) Intercept β 0 Regresson lne Slope β 1 s negatve x
Smple Lnear Regresson Equaton No Relatonshp E(y) Intercept β 0 Regresson lne Slope β 1 s 0 x
Estmated Smple Lnear Regresson Equaton The estmated smple lnear regresson equaton ŷ = b + b x 0 1 The graph s called the estmated regresson lne. b 0 s the y ntercept of the lne. b 1 s the slope of the lne. ŷŷ s the estmated value of y for a gven x value.
Estmaton Process Regresson Model y = β 0 + β 1 x +ε Regresson Equaton E(y) = β 0 + β 1 x Unknown Parameters β 0, β 1 Sample Data: x y x 1 y 1.... x n y n b 0 and b 1 provde estmates of β 0 and β 1 Estmated Regresson Equaton ŷ = b + b x 0 1 Sample Statstcs b 0, b 1
Least Squares Method Least Squares Crteron mn (y y $ ) where: y = observed value of the dependent varable for the th observaton ^ y = estmated value of the dependent varable for the th observaton
Least Squares Method Slope for the Estmated Regresson Equaton b 1 = x y x ( x y ) ( x ) n n
Least Squares Method y-intercept for the Estmated Regresson Equaton b = y bx 0 1 where: x = value of ndependent varable for th observaton y = value of dependent varable for th _ observaton x = mean value for ndependent varable _ y = mean value for dependent varable n = total number of observatons
Smple Lnear Regresson Example: Reed Auto Sales Reed Auto perodcally has a specal week-long sale. As part of the advertsng campagn Reed runs one or more televson commercals durng the weekend precedng the sale. Data from a sample of 5 prevous sales are shown on the next slde.
Smple Lnear Regresson Example: Reed Auto Sales Number of TV Ads 1 3 1 3 Number of Cars Sold 14 4 18 17 7
Estmated Regresson Equaton Slope for the Estmated Regresson Equaton b ( x x )( y y ) 0 = = = 5 ( ) 4 1 x x y-intercept for the Estmated Regresson Equaton b 0 = y b 1 x = 0 5() = 10 Estmated Regresson Equaton yˆ = 10 + 5 x
Scatter Dagram and Trend Lne Cars Sold 30 5 0 15 10 5 0 y = 5x + 10 0 1 3 4 TV Ads
Coeffcent of Determnaton Relatonshp Among SST, SSR, SSE SST = SSR + SSE ( y y ) ( y ˆ y ) = ( y y ˆ ) + where: SST = total sum of squares SSR = sum of squares due to regresson SSE = sum of squares due to error
Coeffcent of Determnaton The coeffcent of determnaton s: r = SSR/SST where: SSR = sum of squares due to regresson SST = total sum of squares
Coeffcent of Determnaton r = SSR/SST = 100/114 =.877 The regresson relatonshp s very strong; 88% of the varablty n the number of cars sold can be explaned by the lnear relatonshp between the number of TV ads and the number of cars sold.
Sample Correlaton Coeffcent r xy = (sgn of b 1 ) Coeffcen t of Determnat on r = (sgn of b ) r xy 1 where: b 1 = the slope of the estmated regresson equaton ˆ = b + b x y 0 1
Sample Correlaton Coeffcent r = (sgn of b ) r xy The sgn of b 1 n the equaton y ˆ = 10 + 5 x s +. 1 r =+.877 r xy r xy = +.9366
Assumptons About the Error Term ε 1. The error ε s a random varable wth mean of zero.. The varance of ε, denoted by σ, s the same for all values of the ndependent varable. 3. The values of ε are ndependent. 4. The error ε s a normally dstrbuted random varable.
Testng for Sgnfcance To test for a sgnfcant regresson relatonshp, we must conduct a hypothess test to determne whether the value of β 1 s zero. Two tests are commonly used: t Test and F Test Both the t test and F test requre an estmate of σ, the varance of ε n the regresson model.
An Estmate of σ Testng for Sgnfcance = = 1 0 ) ( ) ˆ ( SSE x b b y y y = = 1 0 ) ( ) ˆ ( SSE x b b y y y where: s = MSE = SSE/(n ) The mean square error (MSE) provdes the estmate of σ, and the notaton s s also used.
Testng for Sgnfcance An Estmate of σ To estmate σ we take the square root of σ. The resultng s s called the standard error of the estmate. s = MSE = SSE n