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 Celsius

Freezing point 32 0

Boiling point 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 XWhere 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 cells4. 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)

Varying with value of X

Constant

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