SECTION 9.3 Measures of Regression and Prediction Intervals 499 The Coefficient of Determination You already know how to calculate the correlation coefficient r. The square of this coefficient is called the coefficient of determination. It can be shown that the coefficient of determination is equal to the ratio of the explained variation to the total variation. The coefficient of determination r 2 is the ratio of the explained variation to the total variation. That is, r2 = Explained variation Total variation . DEFINITION It is important that you interpret the coefficient of determination correctly. For instance, if the correlation coefficient is r = 0.900, then the coefficient of determination is r2 = 10.90022 = 0.810. This means that 81% of the variation in y can be explained by the relationship between x and y. The remaining 19% of the variation is unexplained and is due to other factors, such as sampling error, coincidence, or lurking variables. Finding the Coefficient of Determination The correlation coefficient for the gross domestic products and carbon dioxide emissions data is r ≈ 0.874. See Example 4 in Section 9.1. Find the coefficient of determination. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation? SOLUTION The coefficient of determination is r2 ≈ 10.87422 ≈ 0.764. Round to three decimal places. Interpretation About 76.4% of the variation in the carbon dioxide emissions can be explained by the relationship between the gross domestic products and carbon dioxide emissions. About 23.6% of the variation is unexplained and is due to other factors, such as sampling error, coincidence, or lurking variables. TRY IT YOURSELF 1 The correlation coefficient for the Old Faithful data is r ≈ 0.979. See Example 5 in Section 9.1. Find the coefficient of determination. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation? Answer: Page A42 EXAMPLE 1 Picturing the World Professor Emeritus Janette Benson (Department of Psychology, University of Denver) performed a study relating the mean age at which infants begin to crawl (in weeks after birth) with the average monthly ambient temperature six months after birth. Her results are based on a sample of 414 infants. Benson believes that the reason for the correlation of temperature and crawling age is that parents tend to bundle infants in more restrictive clothing and blankets during cold months.This bundling does not allow the infant as much opportunity to move and experiment with crawling. 35 28 29 30 31 32 33 34 45 55 65 75 x Temperature (in °F) Crawling age (in weeks) y The correlation coefficient is r ? −0.700. What percent of the variation in the data can be explained? What percent is due to other factors, such as sampling error, coincidence, or lurking variables?
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