"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
29. New Vehicle Sales Construct a 95% prediction interval for new vehicle sales for General Motors in Exercise 19 when the number of new vehicles sold by Ford is 2028 thousand."

Verified step by step guidance

1. Identify the regression model from Exercise 19, which relates new vehicle sales of General Motors (GM) to the number of new vehicles sold by Ford. This model typically has the form: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=bx+a$, where $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ is the predicted GM sales, $x$ is Ford sales, $b$ is the slope, and $a$ is the intercept.
2. Calculate the predicted GM sales ($\mathrm{\backslash (\backslash hat\{y\}\backslash )}$) by substituting $x=\; 2028$ (thousand vehicles sold by Ford) into the regression equation.
3. Determine the standard error of the prediction, which accounts for both the variability of the regression line and the individual prediction. This involves the residual standard error (or standard deviation of errors), the sample size, and the distance of $x$ from the mean of the predictor variable.
4. Find the critical t-value for a 95% prediction interval using the appropriate degrees of freedom (usually $n\; -\; 2$, where $n$ is the sample size). This value comes from the t-distribution table.
5. Construct the 95% prediction interval using the formula: $\backslash (\backslash hat\{y\}\backslash )\; \backslash (\backslash pm\backslash )\; t\_\{\backslash (\backslash alpha\backslash )/2,\; n-2\}\; \backslash (\backslash times\backslash )\; \backslash (\backslash text\{standard\; error\; of\; prediction\}\backslash )$. Interpret this interval as the range in which we expect the actual GM sales to fall with 95% confidence when Ford sells 2028 thousand vehicles.

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 the estimated regression line and the variability of individual data points.

Simple Linear Regression

Simple linear regression models the relationship between two variables by fitting a straight line to observed data. It is used here to predict new vehicle sales for General Motors based on Ford's sales, assuming a linear association between the two.

Confidence Level and Interpretation

The confidence level (e.g., 95%) indicates the probability that the prediction interval contains the true future value. Interpreting the interval means understanding that there is a 95% chance the actual new vehicle sales for General Motors will fall within the calculated range when Ford's sales are 2028 thousand.