500 CHAPTER 9 Correlation and Regression The Standard Error of Estimate When a ny@value is predicted from an x@value, the prediction is a point estimate. You can construct an interval estimate for ny, but first you need to calculate the standard error of estimate. The standard error of estimate se is the standard deviation of the observed yi@values about the predicted ny@value for a given x i@value. It is given by se = CΣ1yi - nyi2 2 n - 2 where n is the number of pairs of data. DEFINITION From this formula, you can see that the standard error of estimate is the square root of the unexplained variation divided by the square root of n - 2. So, the closer the observed y@values are to the predicted ny@values, the smaller the standard error of estimate will be. Finding the Standard Error of Estimate se In Words In Symbols 1. Make a table that includes the five column xi, yi, nyi, 1yi - nyi2, headings shown at the right. 1yi - nyi2 2 2. Use the regression equation to calculate nyi = mxi + b the predicted y@values. 3. Calculate the sum of the squares of the Σ1yi - nyi2 2 differences between each observed y@value and the corresponding predicted y@value. 4. Find the standard error of estimate. se = CΣ1yi - nyi2 2 n - 2 GUIDELINES Instead of the formula used in Step 4, you can also find the standard error of estimate using the alternative formula se = CΣy2 - bΣy - mΣxy n - 2 . You can use this formula when you have already calculated the slope m, the y@intercept b, and several of the sums. For instance, consider the gross domestic products and carbon dioxide emissions data (see Example 4 in Section 9.1 and Example 1 in Section 9.2). To use the alternative formula, note that the regression equation for these data is ny = 187.660x + 44.663 and the values of the sums are Σy2 = 3,548,633.25, Σy = 5269.5, and Σxy = 16,687.99. So, using the alternative formula, the standard error of estimate is se = CΣy2 - bΣy - mΣxy n - 2 = B3,548,633.25 - 144.6632(5269.52 - 1187.6602116,687.992 10 - 2 ≈ 150.671.
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