"In Exercises 19-24, construct the indicated prediction interval and interpret the results.
22. Construct a 95% prediction interval for the fuel efficiency of an automobile in Exercise 12 that has an engine displacement of 265 cubic inches."

Verified step by step guidance

1. Identify the regression equation from Exercise 12, which relates fuel efficiency (dependent variable) to engine displacement (independent variable). This equation typically has the form: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=b\_0\; +b\_1x$, where $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ is the predicted fuel efficiency and $x$ is the engine displacement.
2. Calculate the predicted fuel efficiency for an engine displacement of 265 cubic inches by substituting $x=\; 265$ into the regression equation to find $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$.
3. Determine the standard error of the prediction, which accounts for both the variability of the regression line and the individual prediction. The formula for the standard error of prediction is: $\mathrm{SE\_\{pred\}\; =\; s\; \backslash (\backslash sqrt\{1\; +\; \backslash frac\{1\}\{n\}\backslash )\; +\; \backslash (\backslash frac\{(x\_0\; -\; \backslash bar\{x\}\backslash ))^2\}\{\backslash (\backslash sum\backslash )\; (x\_i\; -\; \backslash (\backslash bar\{x\}\backslash ))^2\}\}}$, where $s$ is the standard error of the estimate, $n$ is the sample size, $x\_0$ is 265, and $\backslash (\backslash bar\{x\}\backslash )$ is the mean of the engine displacements.
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: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}\backslash (\backslash pm\backslash )\; t\_\{\backslash (\backslash alpha\backslash )/2,\; n-2\}\; \backslash (\backslash times\backslash )\; SE\_\{pred\}$. This interval estimates the range in which the fuel efficiency of a single automobile with 265 cubic inches engine displacement is expected to fall with 95% confidence.

Key Concepts

Prediction Interval

A prediction interval estimates the range within which a single new observation is expected to fall, with a certain level of confidence. Unlike confidence intervals for mean responses, prediction intervals account for both the variability in the estimated mean and the individual data point's variability.

Regression Analysis

Regression analysis models the relationship between a dependent variable and one or more independent variables. In this context, it helps predict fuel efficiency based on engine displacement by fitting a line that best describes their relationship.

Confidence Level and Interpretation

The confidence level, such as 95%, indicates the probability that the interval contains the true value for a new observation. Interpreting the prediction interval means understanding that there is a 95% chance the actual fuel efficiency for an engine displacement of 265 cubic inches falls within the calculated range.