SECTION 9.3 Measures of Regression and Prediction Intervals 503 x CO2 emissions (in millions of metric tons) GDP (in trillions of dollars) y y 90% prediction intervals 2 3 4 5 6 −100 100 300 500 700 900 1100 1300 1500 Constructing a Prediction Interval Using the results of Example 2, construct a 90% prediction interval for the carbon dioxide emissions when the gross domestic product is $2.8 trillion. What can you conclude? SOLUTION Because n = 10, there are d.f. = 10 - 2 = 8 degrees of freedom. Using the regression equation ny = 187.660x + 44.663 and x = 2.8 the point estimate is ny = 187.660x + 44.663 = 187.66012.82 + 44.663 = 570.111. From Table 5, the critical value is tc = 1.860 and from Example 2, se ≈ 150.669. From Example 4 in Section 9.1, you found that Σx = 25.7 and Σx2 = 82.81. Also, x = 2.57. Using these values, the margin of error is E = tc seC1 + 1 n + n1x0 - x2 2 nΣx2 - 1Σx22 ≈ 11.86021150.6692C1 + 1 10 + 1012.8 - 2.5722 10182.812 - 125.722 ≈ 294.344. Using ny = 570.111 and E ≈ 294.344, the prediction interval is constructed as shown. Left Endpoint Right Endpoint ny - E ≈ 570.111 - 294.344 ny + E ≈ 570.111 + 294.344 = 275.767 = 864.455 275.767 6 y 6 864.455 Interpretation You can be 90% confident that when the gross domestic product is $2.8 trillion, the carbon dioxide emissions will be between 275.767 and 864.455 million metric tons. TRY IT YOURSELF 3 Using the results of Example 2, construct a 95% prediction interval for the carbon dioxide emissions when the gross domestic product is $4 trillion. What can you conclude? Answer: Page A42 For x-values near x, the prediction interval for y becomes narrower. For x-values further from x, the prediction interval for y becomes wider. (This is one reason why the regression equation should not be used to predict y-values for x-values outside the range of the observed x-values in the data.) For instance, consider the 90% prediction intervals for y in Example 3 shown at the left. The range of the x-values is 0.9 … x … 5.4. Notice how the confidence interval bands curve away from the regression line as x gets closer to 0.9 or to 5.4. EXAMPLE 3
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