The Regression Equation Sections 5.3, 5.4
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1 The Regression Equation Sections 5.3, 5.4 Lecture 13 Robb T. Koether Hampden-Sydney College Fri, Feb 5, 2016 Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
2 Outline 1 Calculating the Regression Line 2 Facts about the Regression Line 3 Assignment Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
3 Outline 1 Calculating the Regression Line 2 Facts about the Regression Line 3 Assignment Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
4 Calculating the Regression Line The Equation of the Regression Line The equation of the regression line is ŷ = a + bx, where and b = r ( sy s x ) a = y bx. Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
5 Example Example (Calculating the Regression Line) Consider again the data x y Find the equation of the regression line. Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
6 Example Example (Calculating the Regression Line) Consider again the data x y Find the equation of the regression line. To do that, we need to find x, y, s x, s y, and r. Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
7 Example Example (Calculating the Regression Line) Using the TI-83, we get x = 8, y = 140, s x = 5, s y = 80, r = Then and b = (0.9375) ( ) 80 = 15 5 a = 140 (15)(8) = 20. Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
8 Example Example (Height vs. Weight) Height (x) Weight (y) Find the equation of the regression line and use it to predict the weight of persons of heights 68, 70, 72, and 84 inches. Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
9 Outline 1 Calculating the Regression Line 2 Facts about the Regression Line 3 Assignment Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
10 Facts about the Regression Line The regression line always passes through the point (x, y). The regression line can be used to predict y given x, but it should not be used to predict x given y. Regressing x on y gives a different line altogether. (Why?) The square of the correlation, r 2, measures the fraction of the variation in y that is explained by the variation in x. Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
11 Example Example (Explaining Variation) In the free-lunch/graduation rate example, r = , so r 2 = We conclude that the variation in the free-lunch rate explains, or accounts for, 73% of the variation in the graduation rate. The remaining 27% is unexplained, or unaccounted for, by the model. Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
12 Example Example (Explaining Variation) In the height/weight example, r = , so r 2 = We conclude that the variation in height explains, or accounts for, 80% of the variation in weight. The remaining 20% is unexplained, or unaccounted for, by the model. Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
13 Outline 1 Calculating the Regression Line 2 Facts about the Regression Line 3 Assignment Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
14 Assignment Assignment Read Sections 5.1, 5.2, 5.3. Apply Your Knowledge: 1, 2, 4, 5. Check Your Skills: 20, 21, 23, 24, 25, 26. Exercises: 30, 34, 35(a-b), 37, 38. Robb T. Koether (Hampden-Sydney College) The Regression EquationSections 5.3, 5.4 Fri, Feb 5, / 13
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