Regression Analysis and Linear Programming Assignment 2, 2018

This document presents a comprehensive solution to a business mathematics assignment, focusing on regression analysis and linear programming. The assignment includes four problems. The first problem explores the relationship between a car's mileage and its sales price using regression analysis, requiring the development of a scatter diagram, interpretation of the relationship, and the development and interpretation of a regression equation. The second problem delves into advertising and its impact on weekly gross revenue, using multiple regression analysis with television and newspaper advertising as independent variables. This involves developing a regression equation, interpreting coefficients, discussing the model's quality, and testing the significance of variables. The third problem utilizes linear programming to maximize a firm's profit by determining the optimal production of two product types, defining the objective function, constraints, and solving the problem using Excel Solver. The final problem employs linear programming to minimize the total cost for a travel company operating two types of vehicles, defining the objective function, constraints, and determining the optimal amount of each vehicle to minimize costs. The assignment provides detailed answers, including Excel outputs and interpretations, and is well-referenced.

University
Business Mathematics
By
Your Name
Date
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© <Your Name> 2018

Assignment 2: Regression Analysis and Linear Programming
Instructions
1. All answers should be presented both in the Word documents and the Excel
spreadsheet
o Word: answer should be after each corresponding question without changing
the order of the questions.
2. Answers should be highlighted in red color.
Problem 1. (20 points)
The cost of a previously owned car depends upon factors such as make and model, model
year, mileage, condition, and whether the car is purchased from a dealer or from a private
seller. To investigate the relationship between the car’s mileage and the sales price, data
were collected on the mileage and the sale price for 10 private sales of model year 2000
Honda Accords (Price Hub website, October 2008).
Miles
(1000s)
Price
($1000s
)
90 7
59 7,5
66 6,6
87 7,2
90 7
106 5,4
94 6,4
57 7
138 5,1
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87 7,2
a. Using Excel and the excel file provided, develop a scatter diagram with miles as the
independent variable.
Answer:
b. What does the scatter diagram developed in part (a) indicate about the relationship
between the two variables?
Answer:
The scatter plot indicates that there is a negative linear relationship between price and
miles. Therefore, the price will reduce with increase in the number of miles covered.
c. Use the least squares method to develop the estimated regression equation.
Answer:
In the least square method, the slope of the regression equation is given by the formula
below [1].
m= N ∑ xy−∑ x ∑ y
N (∑ x2 ) − ( ∑ x )2
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m= 10 ( 5667.7 )−(874)(66.4 )
10 ( 81540 )− ( 874 )2 =−1356.6
51524 =−0.0263
The intercept is given by the equation:
c= ( ∑ y ) ( ∑ x2 )−(∑ x) ( ∑ xy )
n(∑ x2)−¿ ¿
c= 66.4 ( 81540 ) − ( 874 ) ( 5667.7 )
10 ( 81540 ) −8742 = 460686.2
51524 =8.941
The regression equation is therefore:
Y =−0.0263 x +8.941
Where Y is the dependent variable (Price), and x is the independent variable (miles)
d. Provide an interpretation for the slope of the estimated regression equation (the
coefficient of the independent variable).
Answer:
The slope is the coefficient of the independent variable and it indicates the extent with
which the dependent variable affects the independent variable [2]. In this case its -
0.0263 meaning that the independent variable is 0.0263 times the dependent variable
but in the negative direction (Negative relationship). A unit change in the dependent
variable would therefore cause the independent variable to change by 0.0263 but in
negative direction
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Problem 2. Advertising (30 points)
The owner of Showtime Movie Theaters, Inc., would like to estimate weekly gross revenue as
a function of advertising expenditure. Historical data for a sample of eight weeks follow.
Weekly Gross Revenue
($1000s)
Television Advertising
($1000) Newspaper Advertising ($1000)
96 5 1.5
90 2 2
95 4 1.5
92 2.5 2.5
95 3 3.3
94 3.5 2.3
94 2.5 4.2
94 3 2.5
a) Develop an estimated regression equation with both television advertising and
newspaper advertising as the independent variables.
Answer:
From the excel snippet above, the regression equation will be:
Y =2.2901 X1 +1.3010 X2 +83.2301
Where Y in the dependent variable (weekly gross revenue), X1 is the first independent
variable (Television advertising), X2 is the second independent variable (Newspaper
advertising) [3]. The intercept is 83.2301.
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b) Interpret the estimated coefficients
Answer:
The coefficient of television advertising is 2.2901 meaning that for every unit change
in gross revenue, the television advertising will change by a multiple of 2.2901. The
coefficient of newspaper advertising is 1.3010 meaning that a unit change of gross
revenue would cause the newspaper advertising by a multiple of 1.3010. The intercept
is 83.23 and it indicates the value of the gross income if both the television advertising
and newspaper advertising were zero [4].
c) Discuss the quality of the estimated model
Answer:
From the excel snippet above the R square value is 0.91904. This means that 91.904%
of the dependent variable (Gross Income) is explained by the dependent variable
hence the model is a good fit [5].
d) Test for the overall significance of the variables used.
Answer:
From the excel summary output the value of significance F is 0.001865242 and
therefore is less than 0.05. This means that the overall results are statistically
significant [6].
e) Use a significance level of α =0.05 to test the significance of x1. Should x1 be dropped
from the model?
Answer:
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The P-value of the television advertising (P=0.000653232) is below the significance
level of 0.05 and therefore it is statistically significant and should not be eliminated in
the model [3].
f) Use a significance level α =0.05 to test the significance of x2. Should x2 be dropped
from the model?
Answer:
The P-value of the newspaper advertising (P=0.009760798) is below the significance
level of 0.05 and therefore it is statistically significant and should not be eliminated in
the model [3].
Problem 3. (25 points)
A firm can produce 2 types of a certain product, namely basic B and deluxe D. The estimated
profit per unit is $ 10 basic and $ 15 deluxe. The firm has 60 hours of labor available per day
and 100 units of capital. Each unit of basic product B requires 2 hours of labor and 4 units of
capital. Each unit of the de-luxe product D requires 4 hours of labor and 5 units of capital.
Labor and capital cannot be negative.
How much of each type of product should the firm produce in order to maximize profit?
a. Define the objective function for a firm and the decision variables (5 points)
Answer:
Let B = Basic car
Let D =Deluxe car
B and D are the decision variables
The objective function will be:
Z=10 B+15 D
b. Define the constraints for the problem (constraint for labor and constraint for capital)
(10 points)
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Answer:
Constraint for labor is given by:
2 B+4 D ≤ 60
Constraint for capital is given by:
4 B+ 5 D ≤ 100
The non-zero constraints are given by:
B ≥0 , D ≥0
c. Solve the problem with excel solver and find what is the optimal amount of B and D
that maximize the profit and how much the profit is? (10 points)
Answer:
Maximum profit is $266.67 when Basic car is $16.67 and deluxe car is $6.67
Problem 4. (25 points)
A travel company operates two types of vehicle, A and B. Vehicle A can carry 40 passengers
and 30 tons of baggage. Vehicle B can carry 60 passengers but only 15 tons of baggage. The
travel company is contracted to carry at least 960 passengers and 360 tons of baggage per
journey. If a vehicle A costs $1,000 to operate per journey and vehicle B costs $ 1,200 to
operate per journey, what choice of vehicle will minimize the total cost per journey?
a. Define the objective function for a firm and the decision variables (5 points)
Answer:
Let A be vehicle A.
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Let B be vehicle B.
A and B are the decision variables.
The objective function will be:
Z=1000 A +1200 B
b. Define the constraints for the problem (constraint for passengers and baggage) (10
points)
Answer:
The constraint for passengers is:
40 A+60 B ≥ 960
The constraint for the baggage’s is:
30 A +15 B ≥ 360
The non-negative constraints are:
A ≥ 0∧B ≥ 0
c. Solve the problem for excel, which is the optimal amount of A and B that minimize
the costs? (10 points)
Answer:
The minimum cost would be $20400 when the number of vehicle A is 6 and vehicle B
is 12.
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References
[1] A. J. John, W. David and G. J. Davidson, "Statistical Thinking in Business," 2nd ed., Boca
Raton: Chapman & Hall/CRC, 2006.
[2] A. Graham, "Statistics," 2nd ed., London: Hodder Education, 2011.
[3] B. S. Daniel, "Statistics: An Introoduction to Quantitative Economic Research," 4th ed.,
Chicago: Rand McNally, 2006.
[4] C. Thomas, "Explanatory Data Analysis in Business and Economics : An Introduction
Using SPSS, Stata, and Excel," Cham : Springer, 2014.
[5] R. A. David, J. S. Dennis, A. W. Thomas, D. C. Jeffrey and J. C. James, "Statistics for
Business & Economics," 13th ed., Boston: Cengage Learning, 2018.
[6] S. C. John, "Introductory Mathematics & Statistics," 6th ed., North Ryde: N.S.W.
McGraw-Hill Education, 2016.
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