2. Simple Linear Regression and Correlation (Part II) By Asst. Prof. Dr. Min Aung

3. Regression equation Regression equation = Least-squares equation = The best-fitting line along which all sample points are scattering = the straight line whose total squared vertical distance from all scatter points is minimum (Least-squared) Ŷ = a + bX Find A in calculator and press it. Find B in calculator and press it. B = Formula 1 (the first one) and A = Formula 1 (the second one) Ŷ is the point estimate for Y given by the regression equation

4. Regression Line (1) Regression line = Least-squares line Substitute the smallest X value in the regression equation Ŷ = a + bX and compute Ŷ. Then, you get a pair (smallest X, corresponding Ŷ). Substitute the largest X value in the regression equation Ŷ = a + bX and compute Ŷ. Then, you get a pair (largest X, corresponding Ŷ). Plot the two points (smallest X, corresponding Ŷ) and (largest X, corresponding Ŷ). Connect the two points by a straight line segment.

5. Regression Line (2) X Y 2 (4, 3) (2, 1) 4

6. Constant or Y-intercept In the regression equation, A is called constant or slope. A is the value of Ŷ when X = 0. Interpretation of A: If X is 0 unit, the estimated Y is A units. X Y 2 (4, 3) (2, 1) 4 (0, 1) 1 is called the y-intercept of the line Ŷ = 1 + 0.5X. 1 is called the constant of the equation Ŷ = 1 + 0.5X.

7. Regression Coefficient or Slope In the regression equation, B is called Regression Coefficient or Slope. B is the value by which Ŷ increases when X increases by 1 unit. Interpretation of B: If X increases by 1 unit, the estimated Y will increase by B units. X Y 2 (4, 3) (2, 2) 4 0.5 is called the regression coefficient of the equation Ŷ = 1 + 0.5X and slope of the line. 0.5 is called the slope of the line with the equation Ŷ = 1 + 0.5X. 0.5 1 1

8. Interval Estimates Compute S e by Formula 7, then use S e and Formula 4 to compute S b.  : b  tS b, where t is found at t-table, Df = n – 2, two-tailed

9. ANOVA Table An = Analysis, O = of,V = Variance  ANOVA SST = (Total variation of Y values from Ῡ) =  (Y - Ῡ) 2 SSR = (Total variation of Ŷ values from Ῡ) =  (Ŷ - Ῡ) 2 SSE = (Total variation of Y values from Ŷ) =  (Y -Ŷ) 2 Table Structure R E T SSDfMS F Formula 9: Denominator Formula 9: numerator SST - SSR n-1 1 n-2 = SSR = SSE  (n-2) MSR  MSE

10. Three Statistics from ANOVA Table F = MSR  MSE: The larger F is, the more likely is the regression model significant R 2 = SSR  SST : The larger R 2 is, the better can the regression model predict Y values S e =  MSE : The smaller S e is, the more precise is the interval estimates for Y values