Functions and
Mathematical Models
Used to compute
the values on a variable [Y]
given values on
other variables [X1, X2 . . .]
Purposes:
1. Prediction of values of Y given known values
of X
Purposes:
1. Prediction of values of Y given known values
of X
2. Explanation by comparing predicted values of
Y with known values of Y
Purposes:
1. Prediction of values of Y given known values
of X
2. Explanation by comparing predicted values of
Y with known values of Y
3. Estimation of Y: Example: using CPI to
estimate real from nominal values
Purposes:
1. Prediction of values of Y given known values
of X
2. Explanation by comparing predicted values of
Y with known values of Y
3. Estimation of Y: Example: using CPI to
estimate real from nominal values
4. Conversion of values to a different scale:
Example: computing index numbers
Other examples of functions:
• Converting absolute numbers into relative
numbers, e.g., sports injury rate, country HIV
rate
Example: conversion of Celsius to
Fahrenheit
Example: conversion of Celsius to
Fahrenheit
AKA: Night of the Living Dead
Chemistry Lesson
Example: conversion of Celsius to
Fahrenheit
At what temperature Fahrenheit
does water freeze?
Example: conversion of Celsius to
Fahrenheit
At what temperature Fahrenheit
does water freeze?
32 degrees
Example: conversion of Celsius to
Fahrenheit
At what temperature Fahrenheit
does water freeze?
32 degrees
At what temperature Celsius does
water freeze?
Example: conversion of Celsius to
Fahrenheit
At what temperature Fahrenheit does
water freeze?
32 degrees
At what temperature Celsius does water
freeze?
0 degrees
Example: conversion of Celsius to
Fahrenheit
At what temperature Fahrenheit
does water boil?
Example: conversion of Celsius to
Fahrenheit
At what temperature Fahrenheit
does water boil?
212 degrees
Example: conversion of Celsius to
Fahrenheit
At what temperature Fahrenheit
does water boil?
212 degrees
At what temperature Celsius does
water boil?
Example: conversion of Celsius to
Fahrenheit
At what temperature Fahrenheit does
water boil?
212 degrees
At what temperature Celsius does water
boil?
100 degrees
Example: conversion of Celsius to
Fahrenheit
Fahrenheit
Freezing point
Boiling point
Celsius
32
0
212
100
Example: conversion of Celsius to
Fahrenheit
• So, in Fahrenheit, to get from freezing to
boiling takes 180 degrees (212 minus 32)
• In Celsius, it takes 100 degrees (100 minus 0)
• Dividing 180 by 100, we see that every 1
degree Celsius is equivalent to 1.8 degrees
Fahrenheit
Example: conversion of Celsius to
Fahrenheit
Y = 32 + 1.8 X
Where Y = Fahrenheit temperature
x= Celsius temperature
Go to Excel file
1. Enter 4 numbers between 32 and 212 in the
next four cells in the Celsius column
2. Enter the formula for the “Celsius to
Fahrenheit” function in the first empty cell in
the Fahrenheit column
3. Copy the function for the next three cells
4. Create XY scatter diagram (along with me)
Function yields a straight line
• Formula: Y = b + aX
“a” is slope: the amount of change in Y given a
unit change in X
“b” is intercept: the value of Y when X=0
• Alternative arrangement: Y = aX + b
Y = b + aX
[Y = aX + b]
is also called
the regression equation
Function is linear:
Y changes by constant absolute
amount given fixed change in X
Linear functions are one of the two
primary types of functions for this
class.
The other type is exponential
functions.
Linear vs. Exponential
Linear
Exponential
Absolute change Constant
Varying with
value of X
Rate of change
(% change)
Constant
Varying with
value of X
Using linear functions
Using linear functions
• To compute Y: Plug in value of X and do
arithmetic
Using linear functions
• To compute Y: Plug in value of X and do
arithmetic
• To solve for X given Y:
• Y = b + aX
• Y – b = aX
• (Y – b)/a = X
Adding trendline to xy scatter
diagrams:
Open the Minimum Wage Excel file from Lab 8
R2 . . . Coefficient of determination
• Measures degree to which predicted values
match actual values
• How close are the points to the line, on
average?
• Zero means no correspondence between
predicted and actual values
• 1.0 means all predicted values exactly match
actual values or all points are exactly on the
regression line