Question 1
Suppose that you work for a firm that operates stores across multiple markets, and you want to understand the revenues earned by specific store locations. To that end, you collect different pieces of information about each store, and run the following regression:
???????????????? = ??0 + ??1(???????? ???? ??????????) + ??2(???????????????? ???? ?????????????? ?????????? ????????) + ??3(???????????? ????????) In this case:
(i) “Revenue” is the revenue (per employee) per week for each store in thousands of dollars
(ii) “Size of Store” is the area of the store’s floorplan (in square metres).
(iii) “Distance to Nearest Major Road” is the distance (in hundreds of metres) to the nearest 4-lane (or larger) road leading to the store.
(iv) “Market Size” is the number of people, in thousands, who reside in the market served by each store.
The regression results from the data you’ve acquired are displayed below:
Coefficient Standard Error of Coefficient
Intercept 4 1
Size of Store 0.1 0.02
Distance to Nearest Major Road –1 0.25
Market Size 0.5 0.1
The questions related to these results are displayed on the following page.
(a) What is the proper, technical interpretation of the coefficient on “Size of Store” in the multiple regression? (5 marks)
(b) In this multiple regression setting, how would you test the hypothesis that the effect of “Size of Store” on revenues was zero at the 5% level of significance for all markets in which your company operates? (5 marks)
(c) Again, in this multiple regression setting, test (at the 5% level of significance) the hypothesis that the variable “Distance to Nearest Major Road” no effect on a store’s revenues.
(5 marks)
(d) Now suppose that you acquire one other piece of information: the weekly advertising budget (in thousands of dollars) for each particular store. You represent this information with the variable “Advertising Budget”, and your regression is now specified as:
???????????????? = ??0 + ??1(???????? ???? ??????????) + ??2(???????????????? ???? ?????????????? ?????????? ????????)
+ ??3(???????????? ????????) + ??4(?????????????????????? ????????????)
Using this new regression, you obtain the following results:
Coefficient Standard Error of Coefficient
Intercept 4 1
Size of Store 0.01 0.02
Distance to Nearest Major Road –1 0.25
Market Size 0.5 0.1
Advertising Budget 2 0.4
In this case, test the hypothesis that the effect of “Size of Store” on revenues was zero at the 5% level of significance for all markets in which your company operates? (5 marks)
(e) Explain why the coefficient value on “Size of Store” and the hypothesis test on this coefficient is different in part (d) than in part (a) and (b). (10 marks)
Question 2
You’ve been contracted by a computer manufacturer to study the price of laptop computers in the market. You are asked to analyze the determinants of these prices, and so you collect data on the characteristics of various laptops sold by different companies, and use a regression to explore this issue:
?????????? = ??0 + ??1(???????? ???? ????????????) + ??2(?????????????????? ??????????) + ??3(??????) In this case:
(v) “Price” is the price of the laptop (in dollars)
(vi) “Size of screen” is the area, in square centimetres, of the laptop’s screen.
(vii) “Processor Speed” is the computer’s processing speed in GHz. (viii) “RAM” is computer’s memory in GB.
The regression results from the data analyzed by the internal research team are displayed below:
Coefficient Standard Error of the Coefficient
Intercept 500 100
Size of screen –5 2
Processor Speed 200 50
RAM 100 10
Standard Error of the Regression = 150
(a) Provide a definition of the coefficient on “Processor Speed”, and using the 5% level of significance, test the hypothesis that this variable has an effect on the price of the laptop. (10 marks)
(b) Suppose that a new laptop has been made by this company that has: (i) a screen whose area is 120 square centimetres, (ii) a processor speed of 4 GHz, and (iii) 8 GB of RAM. If the company believed that a laptop like this would have a price of $1000, respond to this assertion by using the information from the regression as well as formal hypothesis testing techniques. (10 marks)
(Please see the following page for the next part of this question)
(c) Now suppose that you do some more analysis of the price of laptops by adding some information to the regression used in parts (a) and (b). In particular, you estimate:
?????????? = ??0 + ??1(???????? ???? ????????????) + ??2(?????????????????? ??????????) + ??3(??????) + ??4(????????h??)
In this case, “Weight” is the weight of the laptop (in grams). The results from this regression are displayed below:
Coefficient Standard Error of the Coefficient
Intercept 500 100
Size of screen –1 2
Processor Speed 200 50
RAM 100 10
Weight –2 0.5
Standard Error of the Regression = 100
In this case, the coefficient on “Size of Screen” changed in this regression compared to the regression in part (a). How can you explain this coefficient’s change, while incorporating the correlation between “Size of Screen” and “Weight” into your answer? (10 marks)
(d) Suppose that the company intends to manufacture the same laptop specified in part (b), but has a specific weight in mind. Specifically, the laptop it will make has: (i) a screen whose area is 120 square centimetres, (ii) a processor speed of 4 GHz, (iii) 8 GB of RAM, and (iv) a weight of 400g. If, again, the company believed that a laptop like this would have a price of $1000, respond to this assertion by using the information from the regression as well as formal hypothesis testing techniques. (10 marks)
Question 3
(a) You work for a large company that wants to compare the performance of two sales teams
(we’ll call them “Team A” and “Team B”) working at the firm. To formalize this comparison, you gather a sample of data on the weekly revenue created by each team. You then use this data to estimate the following regression:
?????????????? = ??0 + ??1(???????? ??)
In this case, “Revenue” represents the weekly revenue (in dollars) generated by the sales team, and “Team A” is a dummy variable equal to one if the revenue is created by “Team A”, and zero if it’s created by “Team B”. Your regression results are listed below:
Coefficient Standard Error of Coefficient
Intercept 4000 1000
Team A 800 200
(i) Interpret the meaning of the intercept in the above regression. (5 marks)
(ii) Interpret the meaning of the coefficient on the variable “Team A” in the above regression. (5 marks)
(iii) Use these regression results to determine the average weekly revenue generated by Team A. (5 marks)
(iv) Use your statistical training to rigorously test whether or not the two teams generate similar or different levels of weekly revenue. (5 marks)
(b) Suppose that instead of running the regression listed above, you instead estimate the following regression:
?????????????? = ??0 + ??1(???????? ??)
In this regression, “Revenue” is still defined in the same way as before, but “Team B” is a dummy variable equal to one if the revenue is created Team B, and zero if it was created by Team A. In this case:
(i) Use the estimated coefficients from part (a) to determine the value of the intercept term in this regression (here in part (b)). Interpret the meaning of the intercept in this case. (5 marks)
(ii) Use the estimated coefficients from part (a) to determine the value of the coefficient on the dummy variable “Team B” in this regression (here in part (b)). Interpret the meaning of the coefficient on “Team B” in this case. (5 marks)