"In Exercises 19-24, construct the indicated prediction interval and interpret the results.
23. Construct a 99% prediction interval for the top speed of an electric car in Exercise 17 that takes 5.9 seconds to accelerate from 0 to 60 miles per hour."

Verified step by step guidance

1. Identify the regression model from Exercise 17, which relates the acceleration time (independent variable) to the top speed (dependent variable) of the electric car. This model typically has the form: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=b+mx$, where $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ is the predicted top speed and $x$ is the acceleration time.
2. Calculate the predicted top speed $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ for the given acceleration time of 5.9 seconds by substituting $x=\; 5.9$ 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 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 the given acceleration time (5.9 seconds), and $\backslash (\backslash bar\{x\}\backslash )$ is the mean of the acceleration times in the sample.
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 corresponds to the desired confidence level.
5. Construct the 99% prediction interval using the formula: $\backslash (\backslash hat\{y\}\backslash )\; \backslash (\backslash pm\backslash )\; t\_\{\backslash (\backslash alpha\backslash )/2,\; n-2\}\; \backslash (\backslash times\backslash )\; SE\_\{pred\}$. This interval estimates the range in which the top speed of a new electric car with an acceleration time of 5.9 seconds is expected 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 mean values, prediction intervals account for both the uncertainty in estimating the mean and the variability of individual data points.

Confidence Level

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

Regression and Prediction Using Independent Variables

Prediction intervals often rely on regression models that relate an independent variable (e.g., acceleration time) to a dependent variable (e.g., top speed). Understanding how to use the regression equation to predict values and calculate intervals is essential for interpreting the results.