"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
28. Total Assets Construct a 90% prediction interval for the total assets in federal defined benefit plans in Exercise 18 when the total assets in IRAs are \$6400 billion."

Verified step by step guidance

1. Identify the regression equation from Exercise 18, which relates total assets in federal defined benefit plans (dependent variable) to total assets in IRAs (independent variable). This equation typically has the form: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=b\_0\; +b\_1x$, where $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ is the predicted total assets in federal defined benefit plans and $x$ is the total assets in IRAs.
2. Calculate the predicted value of total assets in federal defined benefit plans by substituting $x=\; 6400$ billion into the regression equation: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=b\_0\; +b\_1\; \backslash (\backslash times\backslash )\; 6400$.
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 the errors), the sample size, and the distance of $6400$ from the mean of the $x$ values. The formula for the standard error of prediction is: $SE\_\{pred\}\; =\; s\; \backslash (\backslash sqrt\{1\; +\; \backslash frac\{1\}\{n\}\backslash )\; +\; \backslash (\backslash frac\{(x\_0\; -\; \backslash bar\{x\}\backslash ))^2\}\{\backslash (\backslash sum\backslash )\; (x\_i\; -\; \backslash (\backslash bar\{x\}\backslash ))^2\}\}$, where $s$ is the residual standard error, $n$ is the sample size, $x\_0\; =\; 6400$, and $\backslash (\backslash bar\{x\}\backslash )$ is the mean of the $x$ values.
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 corresponds to the desired confidence level.
5. Construct the 90% prediction interval using the formula: $\backslash (\backslash hat\{y\}\backslash )\; \backslash (\backslash pm\backslash )\; t^*\; \backslash (\backslash times\backslash )\; SE\_\{pred\}$. Interpret this interval as the range in which we expect the total assets in federal defined benefit plans to fall with 90% confidence when the total assets in IRAs are 6400 billion.

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

Linear Regression and Prediction

Linear regression models the relationship between a dependent variable and one or more independent variables. To construct a prediction interval, the regression equation is used to predict the dependent variable's value for a given independent variable, incorporating the model's error variance.

Confidence Level and Interpretation

The confidence level (e.g., 90%) indicates the probability that the prediction interval contains the true value of a future observation. Interpreting the interval means understanding that, over many samples, 90% of such intervals will capture the actual total assets for the given IRA assets.