SECTION 9.2 Linear Regression 489 Applications of Regression Lines When the correlation between x and y is significant (see Section 9.1), the equation of a regression line can be used to predict y@values for certain x@values. Prediction values are meaningful only for x@values in (or close to) the range of the observed x@values in the data. For instance, in Example 1 the observed x@values in the data range from $0.9 trillion to $5.4 trillion. So, it would not be appropriate to use the regression equation found in Example 1 to predict carbon dioxide emissions for gross domestic products such as $0.2 trillion or $14.5 trillion. To predict y@values, substitute an x@value into the regression equation, then calculate ny, the predicted y@value. This process is shown in the next example. Predicting y-Values Using Regression Equations The regression equation for the gross domestic products (in trillions of dollars) and carbon dioxide emissions (in millions of metric tons) data is ny = 187.660x + 44.663. See Example 1. Use this equation to predict the expected carbon dioxide emissions for each gross domestic product. 1. $1.2 trillion 2. $2.0 trillion 3. $2.6 trillion SOLUTION Recall from Section 9.1, Example 7, that x and y have a significant linear correlation. So, you can use the regression equation to predict y@values. Note that the given gross domestic products are in the range ($0.9 trillion to $5.4 trillion) of the observed x@values. To predict the expected carbon dioxide emissions, substitute each gross domestic product for x in the regression equation. Then calculate ny. 1. ny = 187.660x + 44.663 Interpretation When the gross domestic = 187.66011.22 + 44.663 product is $1.2 trillion, the predicted CO2 = 269.855 emissions are 269.855 million metric tons. 2. ny = 187.660x + 44.663 Interpretation When the gross domestic = 187.66012.02 + 44.663 product is $2.0 trillion, the predicted CO2 = 419.983 emissions are 419.983 million metric tons. 3. ny = 187.660x + 44.663 Interpretation When the gross domestic = 187.66012.62 + 44.663 product is $2.6 trillion, the predicted CO2 = 532.579 emissions are 532.579 million metric tons. TRY IT YOURSELF 3 The regression equation for the Old Faithful data is ny = 12.481x + 33.683. Use this to predict the time until the next eruption for each eruption duration. (Recall from Section 9.1, Example 6, that x and y have a significant linear correlation.) 1. 2 minutes 2. 3.32 minutes Answer: Page A42 When the correlation between x and y is not significant, the best predicted y@value is y, the mean of the y@values in the data. EXAMPLE 3 Picturing the World The scatter plot shows the relationship between the number of farms (in thousands) in a state and the total net income of the farms (in millions of dollars) in that state. (Source: U.S. Department of Agriculture) Net income (in millions of dollars) Farms (in thousands) x y r ≈ 0.570 50 100 150 200 250 2,000 4,000 6,000 8,000 10,000 12,000 Describe the correlation between these two variables. Use the scatter plot to predict the total net income in a state that has 150,000 farms 1x = 1502. The regression line for this scatter plot is ny = 25.696x + 622.707. Use this equation to predict the total net income in a state that has 150,000 farms. (Per the methods of Section 9.1, x and y have a significant linear correlation.) How does your algebraic prediction compare with your graphical one?
RkJQdWJsaXNoZXIy NjM5ODQ=