Question

A large food chain owns a number of pharmacies that operate in a variety of settings. Some are situated in small towns and are open for only 8 hours a day, 5 days per week. Others are located in shopping malls and are open for longer hours. The analysts on the corporate staff would like to develop a model to show how a storeâs revenues depend on the number of hours that it is open. They have collected the following information from a sample of stores.

Hours of Operation Average Revenue ($)
40 5958
44 6662
48 6004
48 6011
60 7250
70 8632
72 6964
90 11097
100 9107
168 11498

Required:
a. Use a linear function to represent the relationship between revenue and operating hours and find the values of the parameters using the LSGRG solver that provide the best fit to the given data. What revenue does your model predict for 120 hours?
b. Suggest a two-parameter nonlinear model for the same relationship and find the parameters using the LSGRG solver that provide the best fit. What revenue does your model predict for 120 hours? Which if the models in (a) and (b) do you prefer and why?

Answer 1

A.)

ŷ = 47.07049X + 4435.08375 ;

10084

B.)

y = - 9964.5212 + 4251.34435In(x) ;

10389

C.)

Logarithmic model

Step-by-step explanation:

Given :

Hours of Operation (X) :

40

44

48

48

60

70

72

90

100

168

Average Revenue Y) :

5958

6662

6004

6011

7250

8632

6964

11097

9107

11498

The best fit Given by a linear model for the data is:

ŷ = 47.07049X + 4435.08375

Average Revenue for 120 hours, X

ŷ = 47.07049(120) + 4435.08375

ŷ = 10083.54255 = 10084

A non-linear model which could be used is a logarithmic model:

General form of a Logarithmic model : y=A+Bln(x)

Equation of best fit :

y = - 9964.5212 + 4251.34435In(x)

Average Revenue for 120 hours, X

y = - 9964.5212 + 4251.34435In(120)

y = - 9964.5212 + 20353.275

y = 10388.754 = 10389

Using the correlation Coefficient value :

Linear mode = 0.8731

Logarithmic model = 0.9084

The logarithmic model is preferred as it has a greater correlation Coefficient value Than the linear model.