10-4 Multiple Regression 553 In 1886, Francis Galton was among the first to study genetics using the methods of regression we are now considering. He wrote the article “Regression Towards Mediocrity in Hereditary Stature,” claiming that heights of offspring regress or revert back toward a mean. Although we continue to use the term “regression,” current applications extend far beyond those involving heights. Finding a Multiple Regression Equation Objective Use sample matched data from three or more variables to find a multiple regression equation that is useful for predicting values of the response variable y. Notation yn = b 0 + b1x1 + b2x2 + g+ bk xk (multiple regression equation found from sample data) y = b0 + b1x1 + b2x2 + g+ bk xk (multiple regression equation for the population of data) yn = predicted value of y (computed using the multiple regression equation) k = number of predictor variables (also called independent variables or x variables) n = sample size (number of values for any one of the variables) Requirements For any specific set of x values, the regression equation is associated with a random error often denoted by e. We assume that such errors are normally distributed with a mean of 0 and a standard deviation of s and that the random errors are independent. Procedure for Finding a Multiple Regression Equation Manual calculations are not practical, so technology must be used. (See the “Tech Center” instructions at the end of this section.) KEY ELEMENTS Predicting Weight EXAMPLE 1 Data Set 1 “Body Data” in Appendix B includes heights (cm), waist circumferences (cm), and weights (kg) from a sample of 153 males. Find the multiple regression equation in which the response variable 1y2 is the weight of a male and the predictor variables are height 1x12 and waist circumference 1x22. SOLUTION Using Statdisk with the sample data in Data Set 1, we obtain the results shown in the display on the top of the next page. The coefficients b0, b1, and b2 are used in the multiple regression equation: yn = -149 + 0.769x 1 + 1.01x2 or Weight = -149 + 0.769 Height + 1.01 Waist The obvious advantage of the second format above is that it is easier to keep track of the roles that the variables play. continued
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