"In Exercises 19-24, construct the indicated prediction interval and interpret the results.
24. Construct a 99% prediction interval for the price of a gas grill in Exercise 18 with a usable cooking area of 900 square inches."

Verified step by step guidance

1. Identify the regression equation from Exercise 18, which relates the price of a gas grill to its usable cooking area. This equation typically has the form: $\mathrm{\backslash hat\{y\}}=b\_0\; +b\_1x$, where $\mathrm{\backslash hat\{y\}}$ is the predicted price and $x$ is the cooking area.
2. Calculate the predicted price $\mathrm{\backslash hat\{y\}}$ for a grill with 900 square inches of cooking area by substituting $x=\; 900$ into the regression equation.
3. Find the standard error of the prediction, which accounts for both the variability in the estimate of the mean response and the variability of individual observations. This involves the residual standard error from the regression and the formula for prediction intervals: $SE\_\{pred\}\; =\; s\; \backslash sqrt\{1\; +\; \backslash frac\{1\}\{n\}\; +\; \backslash frac\{(x\_0\; -\; \backslash bar\{x\})^2\}\{\backslash sum\; (x\_i\; -\; \backslash bar\{x\})^2\}\}$, where $s$ is the standard error of the estimate, $n$ is the sample size, $x\_0$ is 900, and $\backslash bar\{x\}$ is the mean of the cooking areas.
4. Determine the critical t-value for a 99% prediction interval with $n\; -\; 2$ degrees of freedom from the t-distribution table.
5. Construct the prediction interval using the formula: $\backslash hat\{y\}\; \backslash pm\; t^*\; \backslash times\; SE\_\{pred\}$. This interval estimates the range in which the price of a gas grill with 900 square inches of cooking area is likely to fall with 99% confidence.

Key Concepts

Prediction Interval

A prediction interval estimates the range within which a single new 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 observations.

Confidence Level (99%)

The confidence level represents the probability that the interval contains the true value. A 99% confidence level means we are 99% confident that the prediction interval includes the actual price of a gas grill with 900 square inches of cooking area.

Regression and Usable Cooking Area as Predictor

Regression analysis models the relationship between a dependent variable (price) and an independent variable (usable cooking area). Using the regression equation from Exercise 18, the cooking area of 900 square inches is plugged in to predict the price and construct the prediction interval.