Quantitative Reasoning for Management

Question 1 (40 marks)

You work for a major manufacturer of computer printers, and your company is trying to  study the factors that influence the profitability of different printer models.  As such, you collect data on the sales price of a variety of different computer printers, and use a regression to study

the way in which different printer characteristics are related to the price of the printer.  Your data contains information on: the sale price of the printer (in dollars), the number of pages printed per minute, the number of pieces of paper held in the paper tray, and the number of pages the

cartridge can print.  You estimate the following regression:

sale price = β0  + β1 (pages printed per Minute) + β2 (paper Held by Tray)

+ β3 (pages printed by cartridge)

The estimation results from this regression are reported below:

The following questions relate to this regression output.

(a)       Bearing in mind that this is a multiple regression, what is the proper interpretation of the coefficient on “Pages Printed per Minute”? (2 marks)

(b)       In this multiple regression setting, how would you test the hypothesis that the effect of “Paper Held by Tray” on the sale price was zero at the 5% level of significance? (4 marks)

(Please see the following page for the next part of this question)

(c)       Again, in this multiple regression setting, would we conclude that the number of pages printed by a cartridge had an impact on the sale price of the printer?  How would we formally  test this notion at the 5% level of significance? (4 marks)

(d)       Now suppose that you obtain data on the clarity of the printer’s output: you get a new

variable called “Clarity”, that uses a scale from 1 to 100 to measure the clearness of the print (1 is the least-clear print, and 100 is the most-clear print).  The output from your regression that

includes this variable is displayed below:

In this case, test the hypothesis that the effect of “Clearness” on sales price was zero at the 5% level of significance. (5 marks)

(e)       By making an explicit reference to the specific correlation between “Paper Held by Tray” and “Clearness”, explain why the inclusion of the variable “Clearness” altered the magnitude and significance of the coefficient on “Paper Held by Tray”.  (10 marks)

(f)        And now suppose that you obtain data on the overall size of the printer – you know its  volume (in cubic centimeters), which is captured by the variable “Size”.  The output from your regression that includes this variable is displayed below:

In this case, test the hypothesis that the effect of “Size” on sales price was zero at the 5% level of significance. (5 marks)

(g)       By making an explicit reference to the specific correlation between “Size” and “Pages

Printed by Cartridge”, explain why the inclusion of the variable “Size” altered the magnitude and significance of the coefficient on “Pages Printed by Cartridge”.  (10 marks)

Question 2 (35 marks)

You are managing a major jewelry store, and you are about to begin selling a newtype of

engagement ring.  To determine the appropriate selling price of the ring, you collect data on the price of a series of different rings on the market, and your data set is composed of the following variables:

(i)        Price: The sale price of the ring (in dollars)

(ii)       Carat: The total number of carats in the ring’s diamond(s)

(iii)      Clarity: The clarity of the diamond (ranked from 1 to 10 on a numerical scale, where

10 is most clear, and 1 is least clear)

(iv)      Gold: This is a dummy variable equal to one if the ring’s band is made of gold, and

zero otherwise.

The results from the regression are displayed below:

The following questions relate to this information.

(a)       Suppose that the store’s owner wants to only make rings that have gold bands, because she believes that they will sell for a higher price than rings whose bands are made of different

materials.  Test this hypothesis at 5% level of significance, and inform the store’s owner if their view on gold bands is supported by the data. (5 marks)

(b)       Your store’s owner wants to begin selling a new ring with the following specification: a 1.5 carat diamond whose clarity is 10, which is set in a gold band.  She wants to set the price of this ring at $3500; use your regression output to determine whether or not customers will pay

$3500 for aring like this. (10 marks)

(c)       Suppose that you gather one other piece of information about rings sold by other stores:

the quality of their cut.  This quality is measured by a 5-point scale, where a value of 1 represents the worst-quality cut, and 5 represents the best-quality cut.  When you include a variable that

represents this characteristic (which we’ll call “Cut”), you obtain the following results:

In this case, explain why the inclusion of “Cut” altered the coefficient on “Carat”.  (10 marks)

(d)       Suppose that your store’s owner still wants to begin selling a new ring described in part (b), but now she specifies the cut of the diamond for the ring -- the ring will have the following specification: a 1.5 carat diamond whose clarity is  10, which is set in a gold band, and the

diamond will have a cut whose quality is equal to 4 on the quality scale.  If the owner still wants to set the price of this ring at $3500, use your regression output in part (c) to determine whether  or not customers will pay $3500 for aring like this.  (10 marks)

Question 3 (25 marks)

(a)       You run a grocery store and you are studying the relative productivity of cashiers

employed in your store.  You believe that cashiers who were hired before the pandemic are more productive than those hired after the pandemic, and you measure productivity by the number of

groceries processed per minute.  In order to study this issue, you use the following regression

GToceTies peT Minute = β0  + β1 (BefoTe pandemic)

In this case:

(i)        “Groceries Per Minute” represents the number of groceries a cashier scans each minute.

(ii)       “Before Pandemic” is a dummy variable equal to one if the cashier was hired prior to  the pandemic began, and zero if the cashier was hired after the pandemic commenced.

Your regression results are listed below:

(i)        Interpret the meaning of the intercept in the above regression.  (3 marks)

(ii)       Interpret the meaning of the coefficient on the variable “Before Pandemic” in the above regression.  (4 marks)

(iii)      Use these regression results to determine the average number of groceries processed

by a cashier who was hired at your store before the pandemic began.  (4 marks)

(iv)      Use your statistical training to rigorously test whether or not cashiers hired prior to

the pandemic are more productive than cashiers who were hired after the pandemic began.  (4 marks)