Question

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
27. Natural Gas Construct a 95% prediction interval for the export of natural gas from the United States in Exercise 17 when the marketed production of natural gas in the United States is 31 trillion cubic feet."

Accepted Answer

Identify the regression equation from Exercise 17, which relates the marketed production of natural gas (independent variable, x) to the export of natural gas (dependent variable, y). This equation typically has the form: \hat{y} = $$b_0$$ + $$b_1 x$$, where $$b_0 is $$the intercept and $$b_1 is $$the slope. Calculate the predicted export value \hat{y} by substituting x = 31 trillion cubic feet into the regression equation. Determine the standard error of the prediction, which accounts for both the variability of the estimate of the mean response and the variability of individual observations. The formula for the standard error of prediction at a specific $$x_0 is$$: SE_{pred} = s \sqrt{1 + \frac{1}{n} + \frac{($$x_0$$ - \bar{x}$$)^2$$}{\sum (x_i - \bar{x}$$)^2$$}}, where s is the standard error of the estimate, n is the sample size, \bar{x} is the mean of the x-values, and $$x_0 = 31. $$Find the critical t-value for a 95% prediction interval with degrees of freedom equal to n - 2. This value comes from the t-distribution table and depends on the confidence level and sample size. Construct the 95% prediction interval using the formula: \hat{y} \pm t_{\alpha/2, n-2} 	imes SE_{pred}. Finally, interpret this interval as the range in which we expect the export of natural gas to fall for a marketed production of 31 trillion cubic feet, with 95% confidence.

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