In Problem 2.4 you were asked to compute a $95 %$ CI on mean gasoline prediction interval
Question:
In Problem 2.4 you were asked to compute a $95 %$ CI on mean gasoline prediction interval on mileage when the engine displacement $x_{1}=275$ in. $^{3}$ Compare the lengths of these intervals to the lengths of the confidence and prediction intervals from Problem 3.5 above. Does this tell you anything about the benefits of adding $x_{6}$ to the model?
Data From Problem 2.4
Table B. 3 presents data on the gasoline mileage performance of 32 different automobiles.
a. Fit a simple linear regression model relating gasoline mileage $y$ (miles per gallon) to engine displacement $x_{1}$ (cubic inches).
b. Construct the analysis-of-variance table and test for significance of regression.
c. What percent of the total variability in gasoline mileage is accounted for by the linear relationship with engine displacement?
d. Find a $95 % \mathrm{CI}$ on the mean gasoline mileage if the engine displacement is 275 in. $^{3}$
e. Suppose that we wish to predict the gasoline mileage obtained from a car with a 275 -in. ${ }^{3}$ engine. Give a point estimate of mileage. Find a $95 %$ prediction interval on the mileage.
f. Compare the two intervals obtained in parts $d$ and e. Explain the difference between them. Which one is wider, and why?
Data From Problem 3.5
Consider the gasoline mileage data in Table B.3.
a. Fit a multiple linear regression model relatmg gasoline mileage $y$ (miles per gallon) to engine displacement $x_{1}$ and the number of carburetor barrels $x_{6}$.
b. Construct the analysis-of-variance table and test for significance of regression.
c. Calculate $R^{2}$ and $R_{\text {Adj }}^{2}$ for this model. Compare this to the $R^{2}$ and the $R_{\text {Adj }}^{2}$ for the simple linear regression model relating mileage to engine displacement in Problem 2.4.
d. Find a $95 %$ CI for $\beta_{1}$.
e. Compute the $t$ statistics for testing $H_{0}$ : $\beta_{1}=0$ and $H_{0}$ : $\beta_{6}=0$. What conclusions can you draw?
f. Find a $95 %$ CI on the mean gasoline mileage when $x_{1}=275$ in. $^{3}$ and $x_{6}=2$ barrels.
g. Find a $95 %$ prediction interval for a new observation on gasoline mileage when $x_{1}=275$ in. ${ }^{3}$ and $x_{6}=2$ barrels.
Step by Step Answer:
Introduction To Linear Regression Analysis
ISBN: 9781119578727
6th Edition
Authors: Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining
