BMAN10960
Quantitative Methods for
Business and Management 2
Lecture 1
Modelling Relationships and Linear
Functions
Overview
Understand linear functions
Solve linear functions
Check your understanding
Limitations of linear functions
Example of a simple function
Our company accountants have informed us that the final cost of our
product depends on the sum of two costs:
1. Unit cost of production of £24 per item.
2. Annual fixed costs of £500 per year for running the
machinery.
The total annual cost (c) for producing (q) number of units
can be expressed using the function:
Total annual costs = 24q + 500
This simple function is a linear function and it is written in the
general form as: y = ax + b
Linear functions
For any linear function expressed as y = ax + b
• a is the slope or gradient of the function (i.e. the slope of the line on a graph)
• b is the intercept on the vertical (y) axis
Cost (C)
e.g. C = 24q + 500
a = 24
b = 500
Drawing/Defining a linear function
Given a straight line (i.e. y = ax + b),
what is the value of a and b?
The intercept, b
b=1
The slope, a is the change
in y / change in x)
Change in
y
Change
in x
a = 4-1/3-0 =3/3=1
a=1
y = 1x + 1
Overview
Understand linear functions
Solve linear functions
Check your understanding
Limitations of linear functions
Solving linear functions
To solve a linear function, we need only TWO sets of co-ordinates or
data points.
If we have TWO data points then we can find the slope and intercept
by solving simultaneous equations.
Example:
At a price of £5 consumers purchase 300 items per week
of a certain product but at a price of £10 consumers only
purchase 200 items per week.
Assume that demand (D) is linearly related to price (P):
D = aP + b
Solving linear functions
Example: At a price of £5 consumers purchase 300 items per week of a
certain product but at a price of £10 consumers only purchase 200 items per
week.
Solution - mathematically:
D(items)
300
Assume that demand (D) is linearly
related
to price (P):
200
100
D = aP + b
300 = 5a + b……………………(equation 1)
and
200 = 10a + b………….............…(equation 2)
5
10
P
(£)
Solving linear functions
STEP 1:
STEP 2:
(I) 300 = 5a + b
(II) 200 = 10a + b
________________
(I-II) 100 = -5a + 0
Use a = -20 in (I):
100 / -5 = a
a = -20
300 = -100 + b
300=5a + b
300 = -5*20 + b
b = 400
D (items)
D = -20p + 400
300
200
100
D (Items)
5
b = 400
300
a= 300-200 = -20
5-10
200
100
5
10
15
P (£)
10
P (£)
How are such models used?
•
Predict within (interpolate) or outside (extrapolate) the observed data
set.
•
•
e.g. for D = -20p + 400 we can ‘predict’ that if the price (p) is set at £8 we
will have a demand of 240 items and if the price is set at £15 we can
‘predict’ a demand of 100 items.
Describe the relationship
•
e.g. for D = -20p + 400 we can see that for every £1 of change in price, we
would expect a change in quantity of -20 units (slope).
400
D
300
D = -20p + 400
200
100
5
10
p
Overview
Understand linear functions
Solve linear functions
Check your understanding
Limitations of linear functions
Quiz 1
Q1. Data was collected on the weight of a male laboratory rat for the first 25 weeks after its birth.
The linear regression equation is y= 40x+100, where x is number of weeks and y is weight in
grams. What does the y-intercept mean in context of the problem?
•
The predicted weight of the rat at birth
•
The current weight of the rat
•
The average increase in the rat's weight.
Q2. There is a linear relationship between the amount of fat in a sandwich and the amount of calories
in a sandwich: y=217.3+35.2x, where x is fat grams and y is caloric content. What is the estimated
caloric content of a sandwich with 8 grams of fat?
• 217.3 calories
• 281.6 calories
• 498.9 calories
• 1738.4 calories
Q3. y is price of pizza and x is the number of toppings. The linear regression equation for the data is y
= 1.5x + 12. Interpret the slope of the equation.
The slope is (0, 1.5). It means that a pizza with 0 toppings will cost 1.5 pounds.
The slope is 1.5. It means that it costs 1.5 pounds per topping.
There are 1.5 toppings.
There are 1.5 pizzas.
Overview
Understand linear functions
Draw and define linear functions
Solve linear functions
Limitations of linear functions
Limitations of models
•
•
•
Models are only as reliable as the data from which they are built.
Models are typically based on a set of assumptions (e.g. specific type
of model or simplification of the data), which might restrict its use.
It is helpful to gather more data in order to ensure that your linear model
makes correct predictions.
Example:
For D = -20p + 400 we assumed that demand was linearly related to price.
Any problems here?
Scatter Diagrams
•
It is helpful to gather more data in order to ensure that your linear
model makes correct predictions.
Example:
In each of the last 5 years the actual costs (C) of producing a
product at various levels of output (O) have been recorded as:
Year
1
2
3
4
5
Output
50
95
200
120
150
Actual Costs
2500
1800
5500
2800
4500
If cost and output are assumed to be linearly related: C =aO + b
Simple scattergram (X-Y scatter)
Vertical axis
(dependent variable)
Chart title
Costs against Output over the last 5 years
6,000
Scatter points
5,000
Axis title
Cost
4,000
3,000
2,000
Axis scale
1,000
Axis scale
0
0
50
100
150
200
Output
Axis title
If available, mention
units in axes titles too
Horizontal axis
(independent
variable)
250
How do we find the best straight line to fit the linear
function?
Costs against Output over the last 5 years
6,000
5,000
Cost
4,000
3,000
2,000
1,000
0
0
50
100
150
200
Output
 We will learn Linear Regression next week.
250
Quiz 2
Lets try this:
A survey finds that sales of a product priced at £10 resulted in 900 units per week
being sold. When priced at £20, only 800 units per week were sold, and at £30, only
700 units per week were sold.
•
Price (in £)
Units sold per week
10
20
30
900
800
700
Determine slope and intercept of linear line going through the points
Price (in £)
Units sold per week
10
20
30
900
800
700
Название диаграммы
1000
900
800
700
y = ax + b
600
500
400
a=-100/10=-10
800 = -10*20+b => b=1000
300
200
100
0
0
5
10
15
20
25
30
35
Summary
Understand linear functions
Solve linear functions
Check your understanding
Limitations of linear functions
Reading: Dewhurst Sections 8.1 and 8.2 (pp 212-219)
Next week:
Linear function (2):
Line of best fit, Covariance, Correlation and Regression