Refer to the Revenue and Profit data set below. Data are in...

Refer to the Revenue and Profit data set below. Data are in billions of dollars. (a) Use MegaStat or MINITAB to find confidence and prediction intervals for $Y$ using the following set of $x$ values: 1.8, 15, and 30. (b) Report the 95 percent confidence interval and prediction interval for $x=15$. (c) Calculate the 95 percent confidence interval for $\mu_y$ using the appropriate method from Chapter 8. (d) Compare the result from part (c) to the confidence interval you reported in part (b). How are they different? $$ \begin{array}{cc} \hline \text { Revenue and Profit of Entertainment Companies }(\boldsymbol{n}=\mathbf{9}) & \text { Erofit }(\boldsymbol{Y}) \\ \hline \text { Revenue }(\boldsymbol{X}) & -0.020 \\ 1.792 & 1.146 \\ 8.931 & -0.978 \\ 2.446 & -0.162 \\ 1.883 & 0.185 \\ 2.490 & 2.639 \\ 43.877 & 0.155 \\ 1.311 & 1.417 \\ 26.585 & 1.267 \\ 27.061 & \text { Prtainment } \\ \hline \end{array} $$

Refer to the Revenue and Profit data set below. Data are in billions of dollars. (a) Use MegaStat or MINITAB to find confidence and prediction intervals for $Y$ using the following set of $x$ values: 1.8, 15, and 30. (b) Report the 95 percent confidence interval and prediction interval for $x=15$. (c) Calculate the 95 percent confidence interval for $\mu_y$ using the appropriate method from Chapter 8. (d) Compare the result from part (c) to the confidence interval you reported in part (b). How are they different?
$$
\begin{array}{cc}
\hline \text { Revenue and Profit of Entertainment Companies }(\boldsymbol{n}=\mathbf{9}) & \text { Erofit }(\boldsymbol{Y}) \\
\hline \text { Revenue }(\boldsymbol{X}) & -0.020 \\
1.792 & 1.146 \\
8.931 & -0.978 \\
2.446 & -0.162 \\
1.883 & 0.185 \\
2.490 & 2.639 \\
43.877 & 0.155 \\
1.311 & 1.417 \\
26.585 & 1.267 \\
27.061 & \text { Prtainment } \\
\hline
\end{array}
$$

Key Concepts

Simple Linear Regression

This concept involves modeling the relationship between a dependent variable and an independent variable using a linear equation. It is used to describe how the average value of the dependent variable changes with the independent variable, and it forms the basis for making predictions, estimating trends, and conducting inferential statistics in a regression context.

Confidence Interval for the Mean Response

A confidence interval for the mean response estimates the range in which the true mean value of the dependent variable is expected to lie at a given level of the independent variable. It accounts for the uncertainty in the estimation of the regression parameters, providing a measure of precision for the expected average outcome at a particular predictor value.

Prediction Interval

A prediction interval provides a range within which an individual new observation is expected to fall, given a specific value of the predictor. Because it reflects both the uncertainty in the estimated regression line and the variability of individual observations around the line, the prediction interval is typically wider than the confidence interval for the mean response.

Comparison Between Confidence and Prediction Intervals

The key difference between these intervals is that the confidence interval for the mean response focuses on the estimation of the average outcome at a specified value of the predictor, while the prediction interval incorporates additional variability to account for individual prediction errors. This means that the prediction interval is always wider, reflecting the extra uncertainty inherent in predicting a single new observation rather than an average value.

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