"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
24. Trees Construct a 90% prediction interval for the trunk diameter of a tree in Exercise 14 when the height is 80 feet."

Verified step by step guidance

1. Identify the regression equation from Exercise 14, which relates tree height to trunk diameter. This equation typically has the form: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=b+mx$, where $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ is the predicted trunk diameter and $x$ is the height of the tree.
2. Calculate the predicted trunk diameter for a tree height of 80 feet by substituting $x=\; 80$ 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 value 80, and $\backslash (\backslash bar\{x\}\backslash )$ is the mean of the observed heights.
4. Find the critical t-value for a 90% 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 prediction interval using the formula: $\backslash (\backslash hat\{y\}\backslash )\; \backslash (\backslash pm\backslash )\; t\_\{\backslash (\backslash alpha\backslash )/2\}\; \backslash (\backslash times\backslash )\; SE\_\{pred\}$. This interval estimates the range in which the trunk diameter of a single tree with height 80 feet is likely to fall with 90% confidence. Finally, interpret this interval in the context of the problem.

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 data points.

Linear Regression and Prediction

Linear regression models the relationship between an independent variable (e.g., tree height) and a dependent variable (e.g., trunk diameter). Using the regression equation, we can predict the dependent variable's value for a given independent variable and construct intervals around this prediction.

Confidence Level and Interpretation

The confidence level (e.g., 90%) indicates the proportion of such intervals that would contain the true value if the experiment were repeated many times. Interpreting a 90% prediction interval means we are 90% confident the actual trunk diameter for a tree 80 feet tall falls within the calculated range.