Exponential Models

1
Exponential Models

• Presentation 2-9

2
Exponential Models

• Consider the following dataset about the colonial
  population from 1610 to 1670.
• Of course, we start with a scatterplot.

Years since 1610 Population in thousands
0 0.35
10 2.30
20 4.60
30 26.60
40 50.40
50 75.10
60 111.90
3
Exponential Models

• From the scatterplot, we probably suspect it is
  not linear.
• The next slide shows the family of curves we
  watch out for.

4
Exponential Scatterplots
Exponential growth

• Exponential models appear when there is a
  constant multiplier.
• For each unit in the x, you multiply each
  successive y term
• If the multiplier is larger than 1, its growth
• If the multiplier is between 0 and 1, its decay.

Exponential decay
5
Exponential Situations

• Money
• If your savings account earns 3 interest per
  year, each year you could multiply your balance
  by 1.03.
• This is an example of the repeated
  multiplication.
• Bacteria
• Each 42 hours, a bacteria doubles so each 42 hour
  unit of time, you multiply by 2.
• Another example of repeated multiplication.

6
Exponential Transformation

• The trick is to change the data such that it is
  linear.
• This can be done by taking the logarithm.
• Remember properties of logarithms (when you
  multiply powers, you add (linear) exponents
• So, essentially, you change the y values to
  exponents
• Now look at the new scatterplot of (x, log y)

Years Population log (population)
0 0.35 -0.455931956
10 2.3 0.361727836
20 4.6 0.662757832
30 26.6 1.424881637
40 50.4 1.702430536
50 75.1 1.875639937
60 111.9 2.048830087
7
Exponential Transformation

• Now look at the new scatterplot of (x, log y)
• This scatterplot looks more centered about a line
  as opposed to a curve.
• Now that it is linear, we do a least-squares
  regression line on the transformed (x, log y)

8
Exponential Transformation

• The r-squared value looks good (0.9345)
• The regression line (LSRL) is given as
• y 0.0414x 0.1523
• This is not quite right as it should be
• log y 0.0414x 0.1523

9
Exponential Transformation

• Now to solve the equation for y, so we dont have
  to deal with the logarithm.
• Start by exponentiating, then using properties of
  logarithms to solve for y-hat.

10
Exponential Transformation

• This could also be done using ln or natural
  logarithms.
• The only difference would be base e instead of
  base 10.

11
Exponential Transformation

• This concludes the presentation.