Mathematical Modeling
Making Predictions with Data
Introduction to Engineering Design
© 2012 Project Lead The Way, Inc.
Function
A rule that takes an input, transforms it, and
produces a unique output
t
d
• Can be represented by
2
3
5
8
10
• Domain – the set of inputs t ≥ 0
• Range – the set of outputs
d ≥3
Distance, d (ft)
– a table that maps an input to an output
– a graph
– an equation involving two variables
13
18
28
43
53
50
y = 5t + 3
40
30
20
10
0
0
1
2
3
4
5
6
7
time, t (s)
8
9
10
Linear Function
A function that demonstrates a constant
rate of change between two quantities
• Can be represented by a line on a coordinate
grid
• Can be represented by a linear equation
involving two variables
 Distance traveled over time
 Cost based on number of
items purchased
50
40
Cost ($)
• Can represent real-life
situations
60
y = 4.5 x
30
20
10
0
0
1
2
3
4 5 6 7 8 9 10 11 12 13
Number of items, n
Linear Equation
A linear function can be expressed by a
linear equation
• An equation involving two variables
 Independent variable, x
 Horizontal axis
 Vertical axis
• Variables can
represent any two
related quantities
90 y
80 100
70
80
60
50 60
40 40
30
20
20
10 0
0
0
0
Distance
Distance, d (ft)
 Dependent variable, y
y = 5x + 3
5
5
Time, t (s)
10
10
15
15
x
Linear Equation
d
2
3
5
8
10
t
13
18
28
43
53
(5, 28)
30
time, t (s)
• Data is often
collected in tables
• Data is graphed on a
coordinate plane as
ordered pairs
25
(3, 18)
20
(2, 13)
15
10
5
0
0
1
2
3
4
Distance, d (ft)
5
6
Linear Equation
• A line can be drawn through data points
– Line-of-best-fit
– Trendline
 y = mx + b
Rise 5
 m = slope =
= 1=
Run
 b = y-intercept =
Distance, d (ft)
• Slope-intercept form
30
25
y = 5x
5 +3
20
15
1
5
10
5
0
0
1
2
3
Time, t (s)
4
5
6
Function Notation
• Functions are often denoted by letters such
as F, f, G, w, V, etc.
• G(t) represents the output value of G at
the input number t
 Garbage production over time
 t is a member of the Domain
Garbage Production
Garbage Produced per
day (tons)
 t≥0
 G(t) is a member of the Range
 G(t) ≥ 427.92 tons
slope m = 20.05 tons/year
 Garbage production
increases by 20.05 tons/year
1200
1000
G(t) = 20.051t + 427.92
800
600
400
200
0
0
10
20
30
Year (t=0 represents 1970)
40
Function Notation
• Example: d(t) = 5t +3
 Slope, m = 5 ft/s
 The toy car moves 5 feet for every second of time
 y-intercept, b = 3 ft
 The toy car is initially 3 feet from the line at time t = 0
 When will the object be 23
feet from the line?

d(t) = 23 = 5t +3,
t=4
distance, d (ft)
 What is the distance at t = 6 s
 d(6) = 5 · (6) + 3 = 33 ft
50
40
y = 5x + 3
33
30
20
10
0
0
1
2
3
4
4
5
time, t (s)
6
7
8
9 10
Correlation Coefficient, r
• Measure of strength of a linear relation
 -1 ≤ r ≤ 1
 r = ±1 is a perfect correlation
 r = 0 indicates no correlation
• Positive r indicates a direct relationship
 As one variable increases, so does the other
• Negative r indicates an inverse relationship
 As one variable increases, the other decreases
• Strength of relationship
 r > 0.8 is a strong correlation
 r < 0.5 is a weak correlation
Coefficient of Determination, r2
• Measure of how well the line represents the data
 0 ≤ r2 ≤ 1
• Portion of the variance of one variable that is
predictable from the other
 Example: r2 = 0.65, 65% of variation in y is due to x.
The other 35% is due to other variable(s).
• Square of the Correlation Coefficient
Finding Trendlines with Excel
• Create table of data
• Common practice to re-label
years starting with n = 1
• Select data
Sales
Fiscal Year
Year
Sales
Fiscal Year
2003 = 1 (millions
(millions$)
$)
2003
2004
2005
2006
2007
2008
2009
2010
2011
03 1
04 2
2.35
2.35
2.22
2.22
05 3
06
4
07
5
08
6
09
7
10
8
11
9
2.34
2.34
2.54
2.54
2.55
2.55
2.75
2.75
3.11
3.11
3.24
3.24
3.15
3.15
Finding Trendlines with Excel
• Insert Scatterplot
Finding Trendlines with Excel
• Format the Scatterplot
• Select the scatterplot
• Choose the Layout tab
• Chart Title
• Axis Titles
• Gridlines
• Legend (delete)
Finding Trendlines with Excel
• Format the Scatterplot
• Select the scatterplot
• Under Chart Tools
• Choose Format tab
• Select Horizontal (Value) Axis in drop down menu
• Choose Format selection
• Adjust the axis options
• Select Vertical (Value) Axis
• Choose Format selection
• Adjust the axis options
Note that the horizontal
axis was formatted to show
several years in the future.
Finding Trendlines with Excel
• Add Trendline
• Select the scatterplot
• Under Chart Tools
• Choose Layout tab
• In the Analysis panel
• Choose Linear Trendline
• Select Trendline (either within
chart or in Current Selection
panel)
• Forecast
• Display Equation
• Display R-squared value
Making Predictions
• Use the trendline to make predictions
– Function notation
S(t) = 0.1335t+2.0269
where S(t) = projected sales
t = year number (t = calendar year - 2002)
Sales (million $)
Sales Forecast
5
4,5
4
3,5
3
2,5
2
1,5
1
0,5
0
S(t) = 0.1335t + 2.0269
R² = 0.8948
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Year, t (where t = 0 represents 2002)
Making Predictions
• Use the trendline to make predictions
– What is the sales projection for 2015?
t = 2015 – 2002 = 13
S(13) = 0.1335(13)+2.0269 = $3.76 million
5
4,5
4
3,5
3
2,5
2
1,5
1
0,5
0
r2 = 0.89
r = 0.89
= 0.94
S(t) = 0.1335t + 2.0269
R² = 0.8948
2015
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Year, t (where t = 0 represents 2002)
2003
Sales (million $)
Sales Forecast
Making Predictions
• Use the trendline to make predictions
S(t) = 0.1335t + 2.0269
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Year, t (where t = 0 represents 2002)
017
Calendar year = 2017
5
4,5
4
3,5
3
2,5
2
1,5
1
0,5
0
2002
2003
t = calendar year – 2002
15 = calendar year – 2002
Calendar year = 15 + 2002
Sales (million $)
– When will the sales reach $4 million?
S(t) = 0.1335t+ 2.0269
4 = 0.1335t + 2.0269
1.9731
0.1335t = 4 –
2.0269
Sales Forecast
1.9731
t=
= 14.8
0.1335
Say t = 15