Math 280 – Introduction to Probability and Statistics

Module 10 Application Assignment (16 Points)

The points on this assignment total up to 45 points. Your percent score from this assignment will be multiplied by 16 to find your actual assignment grade (out of 16) for the gradebook.

PART I. PRICE ELASTICITY

You are a financial manager for an IT company specializing in uninterruptible power supplies (UPS) and other battery / remote power solutions. Your area Vice President gives you the historical sales data below and asks you to provide a sales forecast for the next four weeks for your company’s best-selling UPS: the Convergys 2.0. [The data set is provided as an Excel file on the assignment page]

		Convergys 2.0
Fiscal Week #		Price ($)	Units Sold
1		367	106
2		200	196
3		372	125
4		376	115
5		521	69
6		480	98
7		319	178
8		371	122
9		600	54
10		289	155
11		569	65
12		321	124
13		249	154
14		435	126
15		247	186
16		317	146
17		573	62
8		624	43
19		437	121
20		442	88
	Convergys 2.0
Fiscal Week #	Price ($)			Units Sold
21	446			115
22	360			109
23	274			127
24	543			65
25	331			146
26	318			174
27	528			85
28	481			88
29	379			153
30	485			100
31	258			176
32	268			151
33	416			129
34	571			56
35	371			124
36	600			51
37	450
38	500
39	525
40	475

1. (4 points) Create a scatterplot for Convergys 2.0, showing the relationship between price and units sold. Be sure the independent and dependent variables are on the correct axes, and that your graph looks professional and is labeled and titled appropriately. Paste your scatterplot below (you may use either MS Excel or Jamovi).

• (6 points) Perform a linear regression on Convergys 2.0’s units sold based on price. Below, write the formula for units sold in terms of price, and provide the coefficient of determination and the coefficient of correlation (make sure you label these and use the right symbols!).

• (4 points) Use your answers from previous questions to predict the number of Convergys 2.0 units that will be sold in weeks 37 through 40 (fill the values in to the table above on page one).

• (3 points) If the price of the Convergys 2.0 increased by $100 from one week to the next, how would you expect the number of units sold to change? Show your work.
  
  

• (6 points) Calculate the residuals for Convergys 2.0’s units sold for weeks 1-36 (round all answers to the nearest unit). Fill-in the residual frequency table below and also include a histogram of this data. Based on this new information, do you still agree that linear regression was a good choice for this analysis? Why? (Hint: how would you describe the shape of this distribution?)
  
  

PART II. DATA CENTER REDESIGN

You are a data center manager and have been asked by your boss to redesign a telecommunications room in one of your smaller facilities. One element of this redesign you would like to optimize is the rack structure – your goal is to provide the greatest amount of equipment storage (by weight) within the cost constraints of the project. There are two types of storage racks you can choose from. Rack A costs $400 per unit, requires three square feet of floor space, and can support up to 80 pounds of equipment. Rack B costs $800 per unit, requires four square feet of floor space, and can support up to 120 pounds of equipment. Your budget for this aspect of the project is $5600, though of course you don’t have to spend all of it. Additionally, you can allocate at most 36 square feet of floor space to the racks in this room. How many of each type of rack should you buy in order to maximize the amount of equipment that could be supported?

1.    (2 points) What are your variables for this project? Assign each a letter and describe them in words.

2.    (2 points) What is the objective function you are being asked to maximize? Assign it a variable, define it in words, and provide an equation for it using the variables you defined above.

3.    (6 points)There are four constraints for this problem (hint: one is for the cost of the racks, one is for floor space, and two other ‘natural’ constraints).  Write these four constraints in equation form

4.    (5 points)Using MS Word, MS Excel, Jamovi or a web-based graphing tool like www.fooplot.com, create a graph of these constraints. Your graph should include the equations of the constraints and indicate which area is the feasible region of possible solutions.

5.    (4 points) Based on your graph, how many points are possible solutions? Give the coordinates of each of these points and evaluate the objective function at each point.

6.    (3 points) Now we can answer our original problem: how many of each type of rack should you buy to maximize the amount of equipment that this room can support? Also, indicate how much of your original budget you will spend and the amount of floor space these racks will take up in total.