530 CHAPTER 10 Correlation and Regression Finding the Equation of the Regression Line Objective Find the equation of a regression line. Notation for the Equation of a Regression Line Sample Statistic Population Parameter y-intercept of regression line b0 b0 Slope of regression line b1 b1 Equation of the regression line yn = b0 + b1x y = b0 + b1x Requirements KEY ELEMENTS 1. The sample of paired 1x, y2 data is a random sample of quantitative data. 2. Visual examination of the scatterplot shows that the points approximate a straight-line pattern.* 3. Outliers can have a strong effect on the regression equation, so remove any outliers if they are known to be errors. Consider the effects of any outliers that are not known errors. In particular, consider the effects of any outliers that dramatically affect the regression line.* *Note: Requirements 2 and 3 above are simplified attempts at checking these formal requirements for regression analysis: • For each fixed value of x, the corresponding values of y have a normal distribution. • For the different fixed values of x, the distributions of the corresponding y-values all have the same standard deviation. (This is violated if part of the scatterplot shows points very close to the regression line while another portion of the scatterplot shows points that are much farther away from the regression line. See the discussion of residual plots in Part 2 of this section.) • For the different fixed values of x, the distributions of the corresponding y values have means that lie along the same straight line. The methods of this section are not seriously affected if departures from normal distributions and equal standard deviations are not too extreme. Formulas for Finding the Slope b1 and y-Intercept b0 in the Regression Equation y n = b0 + b1x FORMULA 10-3 Slope: b1 = r sy sx where r is the linear correlation coefficient, sy is the standard deviation of the y values, and sx is the standard deviation of the x values. FORMULA 10-4 y-intercept: b0 = y - b1x The slope b1 and y-intercept b0 can also be found using the following formulas that are useful for manual calculations or writing computer programs: b1 = n1Σxy2 - 1Σx21Σy2 n1Σx22 - 1Σx22 b0 = 1Σy21Σx22 - 1Σx21Σxy2 n1Σx22 - 1Σx22 Rounding the Slope b1 and the y-Intercept b0 Round b1 and b0 to three significant digits. It’s difficult to provide a simple universal rule for rounding values of b1 and b0, but this rule will work for most situations in this book. (Depending on how you round, this book’s answers to examples and exercises may be slightly different from your answers.)
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