"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
30. New Vehicle Sales Construct a 99% prediction interval for new vehicle sales for Honda in Exercise 20 when the number of new vehicles sold by Toyota is 2159 thousand."

Verified step by step guidance

1. Identify the regression equation from Exercise 20, which relates the number of new vehicles sold by Toyota (independent variable, x) to the number of new vehicles sold by Honda (dependent variable, y). This equation typically has the form: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=b\mathrm{\_0}+b\mathrm{\_1}x$, where $b\mathrm{\_0}$ is the intercept and $b\mathrm{\_1}$ is the slope.
2. Calculate the predicted value $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ for Honda vehicle sales when Toyota sales are 2159 thousand by substituting $x=\; 2159$ 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 $x$ is: $\mathrm{SE\_\{pred\}\; =\; s\; \backslash (\backslash sqrt\{1\; +\; \backslash frac\{1\}\{n\}\backslash )\; +\; \backslash (\backslash frac\{(x\; -\; \backslash bar\{x\}\backslash ))^2\}\{\backslash (\backslash sum\backslash )\; (x\_i\; -\; \backslash (\backslash bar\{x\}\backslash ))^2\}\}}$, where $s$ is the standard error of the estimate, $n$ is the sample size, $\mathrm{\backslash (\backslash bar\{x\}\backslash )}$ is the mean of the $x$ values, and $\mathrm{x\_i}$ are the observed $x$ values.
4. Find the critical t-value for a 99% prediction interval with degrees of freedom equal to $n\; -\; 2$. This value comes from the t-distribution table and reflects the desired confidence level.
5. Construct the 99% prediction interval using the formula: $\mathrm{\backslash (\backslash hat\{y\}\backslash )\; \backslash (\backslash pm\backslash )\; t\_\{\backslash (\backslash alpha\backslash )/2,\; n-2\}\; \backslash (\backslash times\backslash )\; SE\_\{pred\}}$. Interpret this interval as the range in which we expect the actual number of Honda vehicle sales to fall with 99% confidence when Toyota sales are 2159 thousand.

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 mean responses, prediction intervals account for both the uncertainty in estimating the mean and the variability of individual outcomes.

Confidence Level

The confidence level, such as 99%, indicates the probability that the prediction interval contains the true future value. A higher confidence level results in a wider interval, reflecting greater certainty that the interval captures the future observation.

Regression and Using Predictor Variables

Prediction intervals often rely on regression models that relate a dependent variable to one or more predictors. Here, the number of new vehicles sold by Toyota is the predictor used to estimate Honda's sales, and the interval incorporates the variability around this regression estimate.