"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
23. Points Earned Construct a 90% prediction interval for total points earned in Exercise 13 when the number of goals allowed by the team is 140."

Verified step by step guidance

1. Identify the regression model from Exercise 13, which relates the number of goals allowed (independent variable) to total points earned (dependent variable). This model typically has the form: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}=b+mx$, where $\mathrm{\backslash (\backslash hat\{y\}\backslash )}$ is the predicted points, $x$ is the number of goals allowed, $b$ is the intercept, and $m$ is the slope.
2. Calculate the predicted total points earned when the number of goals allowed is 140 by substituting $x=\; 140$ 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 regression line and the individual prediction. This involves the residual standard error (or standard deviation of the residuals), the sample size, and the distance of 140 from the mean of the goals allowed values.
4. Find the critical t-value for a 90% prediction interval using the appropriate degrees of freedom (usually $n\; -\; 2$, where $n$ is the sample size). This t-value corresponds to the desired confidence level and accounts for the uncertainty in the estimate.
5. Construct the 90% prediction interval using the formula: $\mathrm{\backslash (\backslash hat\{y\}\backslash )}\backslash (\backslash pm\backslash )\; t^*\; \backslash (\backslash times\backslash )\; \backslash (\backslash text\{standard\; error\; of\; prediction\}\backslash )$. Interpret this interval as the range in which we expect the total points earned to fall for a team that allows 140 goals, with 90% confidence.

Key Concepts

Prediction Interval

A prediction interval estimates the range within which a single future observation is expected to fall, with a specified level of confidence. Unlike confidence intervals for mean responses, prediction intervals account for both the variability in the estimated regression line and the random error of individual observations.

Linear Regression and Model Interpretation

Linear regression models the relationship between a dependent variable and one or more independent variables. Understanding how to use the regression equation to predict values and interpret coefficients is essential for constructing prediction intervals based on given predictor values.

Confidence Level and Its Role in Interval Estimation

The confidence level (e.g., 90%) indicates the proportion of similarly constructed intervals that would contain the true value in repeated sampling. It reflects the degree of certainty in the interval estimate and affects the width of the prediction interval.