"In Exercises 19-24, construct the indicated prediction interval and interpret the results.
21. Construct a 95% prediction interval for the number of hours of sleep for an adult in Exercise 11 who is 45 years old."

Verified step by step guidance

1. Identify the regression equation from Exercise 11, which relates the number of hours of sleep (dependent variable) to age (independent variable). This equation typically has the form: $ŷ=b0+b1x$, where $ŷ$ is the predicted value, $b\_0$ is the intercept, $b\_1$ is the slope, and $x$ is the age.
2. Calculate the predicted number of hours of sleep for a 45-year-old by substituting $x\; =\; 45$ into the regression equation to find $ŷ$.
3. Determine the standard error of the prediction, which accounts for both the variability of the estimate of the mean response and the individual variability around the regression line. The formula for the standard error of prediction is: $SE\_\{pred\}\; =\; s\; \backslash (\backslash sqrt\{1\; +\; \backslash frac\{1\}\{n\}\backslash )\; +\; \backslash (\backslash frac\{(x\; -\; \backslash bar\{x\}\backslash ))^2\}\{\backslash (\backslash sum\backslash )\; (x\_i\; -\; \backslash (\backslash bar\{x\}\backslash ))^2\}\}$, where $s$ is the standard error of the estimate, $n$ is the sample size, $\backslash (\backslash bar\{x\}\backslash )$ is the mean of the predictor values, and $x$ is the specific value (45 in this case).
4. Find the critical t-value for a 95% prediction interval with $n\; -\; 2$ degrees of freedom from the t-distribution table.
5. Construct the 95% prediction interval using the formula: $ŷ\; \backslash (\backslash pm\backslash )\; t\_\{\backslash (\backslash alpha\backslash )/2,\; n-2\}\; \backslash (\backslash times\backslash )\; SE\_\{pred\}$. This interval estimates the range in which the number of hours of sleep for an individual 45-year-old adult is likely to fall with 95% 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 the mean, prediction intervals account for both the variability in the estimate and the individual data point's variability.

Regression Analysis

Regression analysis models the relationship between a dependent variable and one or more independent variables. In this context, it helps predict the number of hours of sleep based on age, providing the basis for constructing prediction intervals.

Confidence Level (95%)

The confidence level indicates the probability that the interval contains the true value. A 95% confidence level means that if we repeated the sampling many times, 95% of the constructed intervals would contain the actual value of the predicted observation.