Use the results of Problem 17 in Section 14.1 to answer the...

Use the results of Problem 17 in Section 14.1 to answer the following questions: (a) Predict the mean annual ROI of all four-year schools whose 2013 cost is $\$ 180,000$ (b) Construct a $95 \%$ confidence interval for the mean annual ROI found in part (a). (c) Predict the annual ROI of a randomly selected four-year school whose 2013 cost is $\$ 180,000$. (d) Construct a $95 \%$ prediction interval for the annual ROI found in part (c). (e) Explain why the predicted lengths in parts (a) and (c) are the same, yet the intervals constructed in parts (b) and (d) are different.

Use the results of Problem 17 in Section 14.1 to answer the following questions:
(a) Predict the mean annual ROI of all four-year schools whose 2013 cost is $\$ 180,000$
(b) Construct a $95 \%$ confidence interval for the mean annual ROI found in part (a).
(c) Predict the annual ROI of a randomly selected four-year school whose 2013 cost is $\$ 180,000$.
(d) Construct a $95 \%$ prediction interval for the annual ROI found in part (c).
(e) Explain why the predicted lengths in parts (a) and (c) are the same, yet the intervals constructed in parts (b) and (d) are different.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It approximates how the mean of the dependent variable changes with changes in the independent variables by fitting a linear equation to observed data.

Prediction of Mean Response

Predicting the mean response involves using the fitted regression model to estimate the average value of the dependent variable for a given value of the independent variable. This prediction represents the conditional mean outcome and is subject to uncertainty quantified through a standard error and a confidence interval.

Confidence Interval for the Mean

A confidence interval for the mean response gives a range of values within which the true mean outcome is expected to lie with a certain level of confidence (such as 95%). It reflects uncertainty in estimating the average effect across a population, excluding the variability due to individual outcomes.

Prediction Interval for an Individual Observation

A prediction interval provides a range within which a single new observation is expected to fall with a specified probability, such as 95%. This interval is wider than the confidence interval for the mean because it includes not only the uncertainty in estimating the mean response but also the inherent variability of individual observations.

Comparison of Confidence and Prediction Intervals

While the predicted mean response and a single prediction might have the same central value when evaluated at a specific point, their intervals differ in width. The confidence interval for the mean is narrower because it only accounts for uncertainty in estimating the average outcome, whereas the prediction interval is wider as it also captures the additional variability inherent in individual observations.

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