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?

What remedies might be appropriate?

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?

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

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?

Copy and complete: Metals usually form $\dots$ oxides while non-metals form $\dots$ oxides.

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 regression model to predict the price of diamonds included the following predictor variables: the weight of the stone (in carats where 1 carat $=0.2$ grams), the color rating (D, E, F, G, H, or I), and the clarity rating (IF, VVS1, VVS2, VS1, or VS2). (a) Identify the quantitative predictor variable(s). (b) How many indicator variables would be included in the model in order to prevent the least squares estimation from failing? (c) Write the proposed model form for predicting a diamond price.

Use the standard error to construct an approximate prediction interval for $Y$. Based on the width of this prediction interval, would you say the predictions are good enough to have practical value?

A regression model to predict the price of diamonds included the following predictor variables: the weight of the stone (in carats where 1 carat = 0.2 gram), the color rating (D, E, F, G, H, or I), and the clarity rating (IF, VVS1, VVS2, VS1, or VS2).

(a) Identify the quantitative predictor variable(s).

(b) How many indicator variables would be included in the model in order to prevent the least squares estimation from failing?

(c) Write the proposed model form for predicting a diamond price.

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.

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.

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

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.

Use the standard error to construct an approximate prediction interval for $Y$. Based on the width of this prediction interval, would you say the predictions are good enough to have practical value?

Use the standard error to construct an approximate prediction interval for $Y$. Based on the width of this prediction interval, would you say the predictions are good enough to have practical value?