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?

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?

Find (a) the gross profit, (b) the net income before taxes, and (c) the net income after taxes for each firm.

Jill’s Fashions had gross sales of $852,300 last year, with returns of$42,800. The inventory on January 1 was $174,690. A total of$345,790 worth of goods was purchased with freight of $18,107. The inventory on December 31 was$158,200. Wages and salaries were $168,240; rent was$48,200; advertising was $24,300; utilities were$11,600; taxes on inventory and payroll totaled $13,880; miscellaneous expenses totaled$21,900; and income taxes were $34,800. Complete the following income statement.

$$\begin{array}{|l l l l|} \hline \text{Jill's Fashions} & \text{Income Statement} & \text{Year Ending December 31}\\ \text{Gross Sales} & & & \underline{~~~~~~~~~~~~~~~~~}\\ \text{Returns} & & & \underline{~~~~~~~~~~~~~~~~~}\\ \text{Net Sales} & & & ~~~~~~~~~~\underline{~~~~~~~~~~~~~~~~~}\\ ~~~~~~~~\text{Inventory, January 1} & & \underline{~~~~~~~~~~~~~~~~~}\\ \text{Cost of Goods} & & &\\ ~~~~~~~~~~~~~~\text{Purchased} & \underline{~~~~~~~~~~~~~~~~~}\\ ~~~~~~~~\text{Freight} & \underline{~~~~~~~~~~~~~~~~~}\\[10pt] ~~~~~~~~\text{Total Cost of Goods Purchased} & &\underline{~~~~~~~~~~~~~~~~~}\\ ~~~~~~~~\text{Total of Goods Available for Sale} & &\underline{~~~~~~~~~~~~~~~~~}\\ ~~~~~~~~\text{Inventory, December 31} & &\underline{~~~~~~~~~~~~~~~~~}\\ \text{Cost of Goods Sold} & & &\underline{~~~~~~~~~~~~~~~~~}\\ \text{Gross Profit} & & &~~~~~~~~~\underline{~~~~~~~~~~~~~~~~~}\\ \text{Expenses}\\ ~~~~\text{Salaries and Wages} & \underline{~~~~~~~~~~~~~~~~~}\\ ~~~~\text{Rent} & \underline{~~~~~~~~~~~~~~~~~}\\ ~~~~\text{Advertising} & \underline{~~~~~~~~~~~~~~~~~}\\ ~~~~\text{Utilities} & \underline{~~~~~~~~~~~~~~~~~}\\ ~~~~\text{Taxes on Inventory, Payroll} & \underline{~~~~~~~~~~~~~~~~~}\\ ~~~~\text{Miscellaneous Expenses} & \underline{~~~~~~~~~~~~~~~~~}\\ ~~~~\text{Total Expenses} & &\underline{~~~~~~~~~~~~~~~~~}\\ \text{Net Income before Taxes} & & &~~~~~~~~~\underline{~~~~~~~~~~~~~~~~~}\\ ~~~~\text{Income Taxes} & & &~~~~~~~~~\underline{~~~~~~~~~~~~~~~~~}\\ \text{Net Income} & & &~~~~~~~~~\underline{~~~~~~~~~~~~~~~~~}\\ \hline \end{array}$$

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) Calculate the t statistic for each coefficient to test for $\beta=0$. (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 the p-value for each coefficient.

$$\begin{array}{lcc} \hline \text { Predictor } & \text { Coefficient } & \text { SE } \\ \hline \text { Intercept } & 1225.4 & 397.3 \\ \text { FloorSpace } & 11.522 & 1.330 \\ \text { CompetingAds } & -6.935 & 3.905 \\ \text { Price } & -0.14955 & 0.08927 \\ \hline \end{array}$$

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 $R_{\text {adj}}$ showing your formulas clearly.

$$\begin{array}{lrcr} \hline \text { Source } & \text { d.f. } & \text { SS } & {\text { MS }} \\ \hline \text { Regression } & 3 & 1,196,410 & 398,803 \\ \text { Error } & 26 & 379,332 & 14,590 \\ \quad \text { Total } & 29 & 1,575,742 & \\ \hline \end{array}$$

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.

Refer to the ANOVA table for this regression.

(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 $R_{\mathrm{adj}}^2$, showing your formulas clearly.

$$\begin{array}{lccr} \hline \text { Source } & \text { d.f. } & \text { SS } & \text { MS } \\ \hline \text { Regression } & 4 & 259,412 & 64,853 \\ \text { Error } & 45 & 224,539 & 4,990 \\ \quad \text { Total } & 49 & 483,951 & \\ \hline \end{array}$$

Refer to the ANOVA table for this regression.

(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 $R_{\mathrm{adj}}^2$, showing your formulas clearly.

$$\begin{array}{lrrr} \hline \text { Source } & \text { d.f. } & {\text { SS }} & {\text { MS }} \\ \hline \text { Regression } & 3 & 1,196,410 & 398,803 \\ \text { Error } & 26 & 379,332 & 14,590 \\ \quad \text { Total } & 29 & 1,575,742 & \\ \hline \end{array}$$

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.

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.

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?

Vail Resorts asked a random sample of guests to rate their satisfaction on various attributes of their visit on a scale of 1–5 with 1 = very unsatisfied and 5 = very satisfied. The estimated regression model was Y = overall satisfaction score, $X_1$ = lift line wait, $X_2$ = amount of ski trail grooming, $X_3$ = safety patrol visibility, and $X_4$ = friendliness of guest services.

(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 Overall Satisfaction when a guest’s satisfaction in all four areas is rated a 5.