CASE 1 – Abigail Faith | Complete Solution

CASE 1 – Abigail Faith

This problem is one which I worked on as a consultant. Nothing is changed except the name of the CEO and the company. 

Abigail Faith is the CEO of AFB Company, a purveyor of computer-aided design software. Abigail has instructed her distributor that her software should be licensed to industrial customers at annual fees, F,  which should depend principally on two  factors - the number of designers, D, and the number of work stations, W, at each company, according to the following relation:

F = X DY WZ, where X = 1000, Y = .55, and Z = .15

While Abigail allows her distributors some latitude in pricing in order to "get the sale," she's concerned that her instructions may not have been taken seriously. Based on the following data (random sample of 50 clients), the question is whether there is evidence that Abigail's guidelines are not being observed.

Client
    Designers
 (D)    Work Stations (W)    Annual Fee (F)
1    50    10    $12,146
2    394    172    $21,030
3    535    288    $21,659
4    236    13    $14,951
5    266    264    $32,021
6    363    148    $23,086
7    244    17    $25,228
8    182    116    $18,575
9    543    525    $38,066
10    87    72    $9,042
11    891    51    $20,010
12    561    499    $52,978
13    978    790    $40,364
14    821    469    $45,340
15    675    582    $61,629
16    863    779    $30,522
17    612    456    $48,157
18    884    275    $36,763
19    75    35    $10,422
20    517    76    $15,125
21    192    173    $11,836
22    324    29    $23,531
23    55    52    $7,798
24    787    111    $27,471
25    148    107    $11,321
26    737    221    $25,090
27    491    209    $42,666
28    654    519    $25,107
29    572    534    $29,566
30    229    86    $18,692
31    355    249    $13,875
32    135    73    $20,253
33    989    47    $44,927
34    248    121    $15,217
35    595    450    $18,958
36    762    547    $60,994
37    282    210    $14,531
38    411    190    $69,682
39    574    556    $17,217
40    377    348    $35,763
41    688    56    $34,403
42    806    431    $70,869
43    363    260    $17,457
44    160    64    $8,240
45    414    357    $40,521
46    351    345    $14,173
47    961    730    $77,115
48    782    748    $46,472
49    612    320    $77,188
50    159    62    $17,090

(The data can be copied and pasted into Excel or SPSS)

The first thing one needs to do is to take the logarithm of both sides (either log base 10 [“log”] or log base e [“ln” – called “natural log”]). This will result in a linear (multiple regression) model, with variables (logF, logD, logW) or (LnF, LnD, LnW). All hypothesis tests should use  = .05.

a) Test whether there is evidence that Abigail’s distributors are violating her edict that X = 1000.
b) Test whether there is evidence that Abigail’s distributors are violating her edict that Y = .55.
c) Test whether there is evidence that Abigail’s distributors are violating her edict that Z = .15.
    
d) What is your summary conclusion after the 3 above analyses? 

e) Test (jointly) whether there is evidence that Abigail’s distributors are violating her edict. 

IN NOTES - 6
Students comprising firm A in a computerized marketing game have approached you for assistance in analyzing the relation between promotional expenditures (X) and demand for their firm’s product (Y) in the firm’s home territory. They believe that the following characteristics hold in this relation: (1) demand in the home territory is affected primarily by promotional expenditures, (2) the relation is either quadratic or linear within the range of x levels of interest to the firm. The team has provided the data shown below for the 14 periods covered in the game to date (X in thousand dollars, Y in thousand units), and has stated that these data span the X levels of interest.
    i:     1               2               3               4              5             6             7
    Xi:  17            15             25             10            18            15           20
    Yi:  56.15     54.50        55.27        52.54       56.23       55.97     55.55
    i:      8               9             10            11            12             13            14 
    Xi:   25            17            13            20            23             25            16
    Yi: 54.32       55.14      54.28       55.78        55.65        54.96        55.06