"In Exercises 19-24, construct the indicated prediction interval and interpret the results.
20. Construct a 90% prediction interval for the average time adults ages 35 to 44 spend per day watching television in Exercise 10 when the average time adults ages 25 to 34 spend per day watching television is 2.25 hours."

Verified step by step guidance

1. Identify the regression equation from Exercise 10, which relates the average time adults ages 25 to 34 spend watching television (independent variable, x) to the average time adults ages 35 to 44 spend watching television (dependent variable, y). This equation typically has the form: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=b\_0\; +b\_1x$.
2. Calculate the predicted average time for adults ages 35 to 44 by substituting $x=\; 2.25$ hours into the regression equation to find $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$.
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 around the regression line. This involves using the formula for the prediction interval standard error: $SE\_\{pred\}\; =\; s\; \backslash (\backslash sqrt\{1\; +\; \backslash frac\{1\}\{n\}\backslash )\; +\; \backslash (\backslash frac\{(x\; -\; \backslash bar\{x\}\backslash ))^2\}\{\backslash (\backslash sum\backslash )\; (x\_i\; -\; \backslash (\backslash bar\{x\}\backslash ))^2\}\}$, where $s$ is the standard error of the estimate, $n$ is the sample size, $\backslash (\backslash bar\{x\}\backslash )$ is the mean of the x-values, and $x$ is 2.25.
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 90% 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 an individual adult aged 35 to 44 is expected to spend watching television, given the average time for adults aged 25 to 34 is 2.25 hours.

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 means, prediction intervals account for both the variability in the sample mean and the individual data points, making them wider and more appropriate for predicting individual outcomes.

Confidence Level

The confidence level, such as 90%, represents the probability that the interval constructed from sample data contains the true value of the parameter or future observation. A 90% prediction interval means we are 90% confident that the new observation will lie within the calculated range.

Regression and Prediction Using Independent Variables

When predicting a response variable based on an independent variable, regression analysis models the relationship between them. Using the regression equation, we can estimate the expected value and construct prediction intervals for new observations at given predictor values, accounting for uncertainty in the estimate.