"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."

Verified step by step guidance

1. 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: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=b\_0\; +b\_1x$, where $b\_0$ is the intercept and $b\_1$ is the slope.
2. Calculate the predicted export value $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ by substituting $x\; =\; 31$ trillion cubic feet into the regression equation.
3. 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: $SE\_\{pred\}\; =\; s\; \backslash (\backslash sqrt\{1\; +\; \backslash frac\{1\}\{n\}\backslash )\; +\; \backslash (\backslash frac\{(x\_0\; -\; \backslash bar\{x\}\backslash ))^2\}\{\backslash (\backslash sum\backslash )\; (x\_i\; -\; \backslash (\backslash bar\{x\}\backslash ))^2\}\}$, where $s$ is the standard error of the estimate, $n$ is the sample size, $\backslash (\backslash bar\{x\}\backslash )$ is the mean of the x-values, and $x\_0\; =\; 31$.
4. 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.
5. Construct the 95% prediction interval using the formula: $\backslash (\backslash hat\{y\}\backslash )\; \backslash (\backslash pm\backslash )\; t\_\{\backslash (\backslash alpha\backslash )/2,\; n-2\}\; \backslash (\backslash times\backslash )\; SE\_\{pred\}$. 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.

Key Concepts

Prediction Interval

A prediction interval estimates the range within which a single future observation is expected to fall, with a specified level of confidence. Unlike confidence intervals for the mean, prediction intervals account for both the uncertainty in estimating the mean and the variability of individual data points.

Linear Regression and Model Use

Prediction intervals often rely on a regression model that relates an independent variable (e.g., marketed production) to a dependent variable (e.g., natural gas export). Understanding how to use the regression equation and its residual variance is essential to calculate the interval for a new input value.

Confidence Level and Interpretation

The confidence level (e.g., 95%) indicates the probability that the prediction interval contains the true future observation. Interpreting this correctly means recognizing that if many such intervals were constructed, about 95% would capture the actual export value for the given production level.