The Linear Regression Model Y Y=βo+BX+c: Observed Value of Y for Xi Predicted Value Random Error of Y for Xi for this X;value Intercept=Bo X X
The Linear Regression Model Random Error for this Xi value Y X Observed Value of Y for Xi Predicted Value of Y for Xi i 0 1 i i Y β β X ε Xi Slope = β1 Intercept = β 0 εi
Linear Regression Equation The simple linear regression equation provides an estimate of the population regression line Estimated (or predicted)Y Estimate of the Estimate of the value for regression regression slope observation i intercept Value of X for Y;=bo+biXi observation i
Linear Regression Equation Yi b 0 b 1 X i ˆ The simple linear regression equation provides an estimate of the population regression line Estimate of the regression intercept Estimate of the regression slope Estimated (or predicted) Y value for observation i Value of X for observation i
The Least Squares Method bo and b are obtained by finding the values of bo and b that minimize the sum of the squared differences between Y andY: 0=min∑(Y,-,)2=min∑(Y,-b+b,X)》2 2-0； a2=0 abo ab
The Least Squares Method b 0 and b1 are obtained by finding the values of b 0 and b1 that minimize the sum of the squared differences between Y and : 2 2 i i i 0 1i Q min (Y Y ) min (Y (b b X )) ˆ Yˆ 0 0 Q b ； 1 0 Q b
0=-2∑0y-b,-bx)=0 ab。 0=2∑y-4-6x=0 bo=y-bx ∑(x-xy-) b1= ∑x-x
0 1 1 2 ( )( ) ( ) i i i i i b y bx x xy y b x x 0 1 0 0 1 1 2 ( )0 2( ) 0 i i i i ii i Q y b bx b Q y b bx x b
y ∑(y-,)2=min 0 x x
。 。 。 。 。 。 。 。 。 。 x y 0 。 。 yi ( ˆ ) min 2 i i y y i y ˆ xi