1. The manager of Collins Import Autos believes the number of cars sold in a day (Q) depends on two factors: (1) the number of hours the dealership is open (H) and (2) the number of salespersons working that day (S). After collecting data for two months (53 days), the manager estimates the following log-linear model:
a. Explain, how to transform the log-linear model into linear form that can be estimated using multiple regression analysis.
The computer output for the multiple regression analysis is shown below:
b. How do you interpret coefficients b and c? If the dealership increases the number of salespersons by 20%, what will be the percentage increase in daily sales?
c. Test the overall model for statistical significance at the 5% significance level.
d. What percent of the total variation in daily auto sales is explained by this equation? What could you suggest to increase this percentage?
e. Test the intercept for statistical significance at the 5% level of significance. If H and S both equal 0, are sales expected to be 0? Explain why or why not?
f. Test the estimated coefficient b for statistical significance. If the dealership decreases its hours of operation by 10%, what is the expected impact on daily sales?
2. Using the optimization theory, analyze the following quotations:
a. The optimal number of traffic deaths in the United States is zero.
b. Any pollution is too much pollution.
c. We cannot pull US troops out of Afghanistan. We have committed so much already.
d. If Congress cuts out the International Space Station (ISS), we will have wasted all of the resources that we have already spent on it. Therefore, we must continue funding the ISS.
e. Since JetGreen Airways has experienced a 25% increase in its insurance premiums, the airline should increase the number of passengers it serves next quarter in order to spread the increase in premiums over a larger number of tickets.
3. You are interviewing three candidates for one sales job position. On the basis of your experience and insight, you believe Jane can sell 600 units a day, Joe can sell 450 units a day, and Joan can sell 400 units a day. The daily salary each person is asking is as follows: Jane $200; Joe $150; and Joan $100. How would you rank the three applicants?
4. Bavarian Crystal Works designs and produces lead crystal wine decanters for export to international markets. The production manager of Bavarian Crystal Works estimates total and marginal production costs to be
and
where costs are measured in U.S. dollars and Q is the number of wine decanters produced annually. Because Bavarian Crystal Works is the only one of many crystal producers in the world market, it can sell as many of the decanters as it wishes for $70 apiece. Total and marginal revenue are TR = 70Q and MR = 70 where revenues are measured in U.S. dollars and Q is the annual decanter production.
a. What is the optimal level of production of wine decanters? What is the marginal revenue from the last wine decanter sold?
b. What are the total revenue, total cost, and net benefit (profit) from selling the optimal number of wine decanters?
c. At the optimal level of production of decanters, an extra decanter can be sold for $70, thereby increasing the total revenue by $70. Why does the manager of this firm not produce and sell one more unit?
5. A decision maker wishes to maximize total benefit, B = 3x + xy + y subject to the cost constraint, C = 4x + 2y = 70. Setup the Lagrangian and then determine the values of x and y at the minimum level of benefit, given the constraint. What are the maximum benefits?