Perform the regression and write the estimated regression equation (round off to 3 or 4 significant digits for clarity). Do the coefficient signs agree with your $a$ priori expectations?

In its recent Fuel Economy Guide, the Environmental Protection Agency (EPA) gives data on 1152 vehicles. There are a number of outliers, mainly vehicles with very poor gas mileage or hybrids with very good gas mileage. If we ignore the outliers, however, the combined city and highway gas mileage of the other 1120 or so vehicles is approximately Normal with mean 18.7 miles per gallon (mpg) and standard deviation 4.3 mpg. How high must a vehicle's gas mileage be in order to fall in the top 10% of the 1120 vehicles?

Using the "Metals" data, construct a correlation matrix of the six independent variables. The response variable is Price/lb. (a) Identify any pairs of independent variables that have a significant pairwise correlation. (b) Using MegaStat or MINITAB, run the regression with all six predictor variables, calculating the VIF for each predictor. (c) Did you see any cause for concern based on the VIF values? Why or why not?

Refrigerator prices are affected by characteristics such as whether or not the refrigerator is on sale, whether or not it is listed as a Sub-Zero brand, the number of doors (one door or two doors), and the placement of the freezer compartment (top, side, or bottom). The table below shows the regression output from a regression model using the natural log of price as the dependent variable. The model was developed by the Bureau of Labor Statistics.

(a) Write the regression model, being careful to exclude the base indicator variable.

(b) Find the $p$-value for each coefficient, using 319 degrees of freedom. Using an $\alpha = .01$, which predictor variable(s) are not significant predictors?

(c) By how much does the natural log of refrigerator price decrease from a two-door, side freezer model to a two-door, top freezer model?

(d) Which model demands a higher price: the side freezer or the one door with freezer model?

How many criterion variables are there in a multiple-regression analysis? How many predictor variables?

Refrigerator prices are affected by characteristics such as whether or not the refrigerator is on sale, whether or not it is listed as a Sub-Zero brand, the number of doors (one door or two doors), and the placement of the freezer compartment (top, side, or bottom). The table below shows the regression output from a regression model using the natural log of price as the dependent variable. The model was developed by the Bureau of Labor Statistics. (a) Write the regression model, being careful to exclude the base indicator variable. (b) Find the p-value for each coefficient, using 319 degrees of freedom. Using an $\alpha=.01$, which predictor variable(s) are not significant predictors? (c) By how much does the natural log of refrigerator price decrease from a two door, side freezer model to a two door, top freezer model? (d) Which model demands a higher price: the side freezer or the one door with freezer model?

$$\begin{array}{lccr} \hline \text { Variable } & \text { Coefficient } & \text { Standard Error } & \boldsymbol{t} \text { Statistic } \\ \hline \text { Intercept } & 5.484092 & 0.13081309 & 41.923 \\ \text { Sale price } & -0.0733 & 0.02338826 & -3.134 \\ \text { Sub-Zero brand } & 1.11962 & 0.14615699 & 7.660 \\ \text { Total capacity (in cubic ft) } & 0.06956 & 0.00535103 & 12.999 \\ \text { Two door, freezer on bottom } & 0.046569 & 0.0808569 & 0.576 \\ \text { Two door, side freezer } & \text { Base } & & \\ \text { Two door, freezer on top } & -0.343246 & 0.03595873 & -9.546 \\ \text { One door with freezer } & -0.709558 & 0.13097047 & -5.418 \\ \text { One door, no freezer } & -0.881981 & 0.14913992 & -5.914 \\ \hline \end{array}$$

Using the "Vail Guest Satisfaction Survey" data, construct a correlation matrix of the 11 independent variables. The response variable is ovalue. (a) Identify the four pairs of independent variables that have the highest pairwise correlation values. Do they show significant correlation? (b) Using MegaStat or MINITAB, run the regression with all 11 predictor variables, calculating the VIF for each predictor. (c) Did you see any cause for concern based on the VIF values? Why or why not?

A regression model to predict Y, the state burglary rate per 100,000 people for 2005 , used the following four state predictors: $X_1$= median age in 2005, $X_2$= number of 2005 bankruptcies, $X_3$= 2004 federal expenditures per capita (a leading predictor), and $X_4$=2005 high school graduation percentage. (a) Calculate the t statistic for each coefficient to test for $\beta_j=0$. (b) Look up the critical value of Student's t in Appendix D for a two-tailed test at $\alpha=.01$. Which coefficients differ significantly from zero? (c) Use Excel to find a p-value for each coefficient.

$$\begin{array}{lrr} \hline \text { Predictor } & \text { Coefficient } & {\text { SE }} \\ \hline \text { Intercept } & 4,198.5808 & 799.3395 \\ \text { AgeMed } & -27.3540 & 12.5687 \\ \text { Bankrupt } & 17.4893 & 12.4033 \\ \text { FedSpend } & -0.0124 & 0.0176 \\ \text { HSGrad\% } & -29.0314 & 7.1268 \\ \hline \end{array}$$

An agribusiness performed a regression of wheat yield (bushels per acre) using observations on 25 test plots with four predictors (rainfall, fertilizer, soil acidity, hours of sun). The standard error was 1.17 bushels. Find the approximate width of a 95 percent prediction interval for wheat yield, assuming that the predictor values for a test plot are near the means of the sample predictors. Would the quick rule give similar results?

A researcher used stepwise regression to create regression models to predict BirthRate (births per 1,000) using five predictors: LifeExp (life expectancy in years), InfMort (infant mortality rate), Density (population density per square kilometer), GDPCap (Gross Domestic Product per capita), and Literate (literacy percent). Interpret these results.

Regression Analysis - Stepwise Selection (best model of each size) 153 observations BirthRate is the dependent variable p-values for the coefficients Nuar 1 2 3 4 5

$$\begin{array}{|c|c|c|c|c|} {\text { LifeExp }} & \text { InfMort } & {\text { Density }} & {\text { GDPCap } \text { Literate }} \\ \hline & .0000 & & & \\ \hline & .0000 & & & .0000 \\ \hline & .0000 & & .0242 & .0000 \\ \hline .5764 & .0000 & & .0311 & .0000 \\ \hline .5937 & .0000 & .6289 & .0440 & .0000 \\ \hline \end{array}$$

$$\begin{array}{ccc}s & \text { Adj R }^2 & R^2 \\ 6.318 & .722 & .724 \\ 5.334 & .802 & .805 \\ 5.261 & .807 & .811 \\ 5.273 & .806 & .812 \\ 5.287 & .805 & .812\end{array}$$

A researcher used stepwise regression to create regression models to predict CarTheft (thefts per 1000) using four predictors: Income (per capita income), Unem (unemployment percent), Pupil/Tea (pupil-to-teacher ratio), and Divorce (divorces per 1,000 population) for the 50 U.S. states. Interpret these results.

$$\begin{aligned} &\begin{array}{|c|c|c|c|c|c|c|c|} \text { Nvar } & {\text { Income }} & {\text { Unem }} & {\text { Pupil/Tea }} & \text { Divorce } & \text { Std. Err } & \text { Adj } R^2 & R^2 \\ \\ 1 & & & .0004 & &167.482 & .218 & .234 \\ 2 & .0018 & & .0000 & & 152.362 & .353 & .379\\ 3 & .0013 & .0157 & .0001 & & 144.451 & .418 & .454 \\ 4 & .0007 & .0323 & .0003 & .1987 & 143.362 & .427 & .474 \\ \end{array} \end{aligned}$$

Observations are taken on sales of a certain mountain bike in 30 sporting goods stores. The regression model was Y= total sales (thousands of dollars), $X_1=$ display floor space (square meters), $X_2=$ competitors" advertising expenditures (thousands of dollars), $X_3=$ advertised price (dollars per unit). (a) Write the fitted regression equation. (b) Interpret each coefficient. (c) Would the intercept seem to have meaning in this regression? (d) Make a prediction for Sales when FloorSpace =80, CompetingAds =100, and Price =1,200.

$$\begin{array}{lc} \hline \text { Predictor } & \text { Coefficient } \\ \hline \text { Intercept } & 1225.44 \\ \text { FloorSpace } & 11.52 \\ \text { CompetingAds } & -6.935 \\ \text { Price } & -0.1496 \\ \hline \end{array}$$

A regression model to predict Y, the state burglary rate per 100,000 people for 2005, used the following four state predictors: $X_1$ = median age in 2005, $X_2$ = number of 2005 bank- ruptcies, $X_3$ = 2004 federal expenditures per capita (a leading predictor), and $X_4$ = 2005 high school graduation percentage.

(a) Calculate the t statistic for each coefficient to test for $\beta_j$ = 0.

(b) Look up the critical value of Student’s t in Appendix D for a two-tailed test at $\alpha$ =.01. Which coefficients differ significantly from zero?

(c) Use Excel to find a $p$-value for each coefficient.

Refer to the ANOVA table below.

(a) State the degrees of freedom for the F test for overall significance.

(b) Use Appendix F to look up the critical value of F for $\alpha$ = .05.

(c) Calculate the F statistic. Is the regression significant overall?

(d) Calculate $R^2$ and $R^2_{adj}$, showing your formulas clearly.