Exponential Regression

1
Exponential Regression

• Section 4.1.1

2
Starter 4.1.1

• The city of Concord was a small town of 10,000
  people in 1950. Returning war veterans and the
  G.I. Bill led to rapid growth which continued
  through the rest of the 20th century. The table
  below shows approximate population figures for
  each decade.
• Use linear regression on your calculator to find
  a mathematical model of the data. HINT Let x
  be years since 1950 instead of calendar years.
  These are called reference years.
• Sketch the scatterplot and LSRL.
• Write the equation and the correlation constant.
  
• Sketch the residual plot and comment on how well
  the LSRL fits the data.

Year 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
Pop 10000 13000 16000 21000 27000 34000 43000 55000 70000 90000 115000
3
Objectives

• Convert exponential data to linear data by use of
  logarithm principles
• Perform linear regression on linearized data
• Evaluate linear fit by using a residual plot
• Convert linear results to an exponential function
  of the form y abx that models the original data

4
Graphing Activity

• Use the special graph paper I give you to graph
  the Concord growth data. Be sure to label axes.
• Put reference year on the x axis
• Put population on the y axis
• What surprising pattern did you find?
• The data are linear
• That wasnt the case in the starter, so why do
  they appear linear now?

5
Review of Logarithms

• To answer the question, we need to remember some
  basic facts about logs
• A logarithm is an exponent
• So when we ask what is the log of 1000, we mean
  what exponent could be put over a base of 10 to
  give a result of 1000?
• 103 1000, so log 1000 3
• Since no base was written, we assume the base is
  10
• If we ask what is log28, the answer is 3 because
  238
• Every exponential statement has an equivalent
  logarithmic statement
• To say 3481 is equivalent to saying log3814
• In general, if abc, then logacb
• Also log10x x and 10log x x logs
  exponents are inverses
• Three important rules govern the arithmetic of
  logs
• Product Rule log(ab) log(a) log(b)
• Quotient Rule log(a/b) log(a) log(b)
• Power Rule log(a)b b log(a)

6
Applying the Rules of Logs

• Consider a general exponential function of the
  form yabx where a and b are unknown constants
• Suppose we take logs of both sides
• log y log (abx)
• log y log(a) log(bx) product rule
• log y log (a) x log(b) power rule
• But a is just an unknown constant, so log(a) is
  also an unknown constant that we could call A
• Similarly, log(b) is a constant that we call B
• So the last line above could be written log y
  A Bx
• But A Bx is just a linear function of x
• So log y is a linear function of x
• Thats why your graph was linear
• Notice that your y axis is scaled in log units,
  so you really graphed x against the log of y, not
  just y itself.

7
Finding the LSRL of Linearized Data

• Return to the starter data and define L3 to be
  log(L2)
• So L3 contains the logs of the y values in L2
• Set up Plot 2 as a scatterplot of L1 L3
• Turn off Y1 and Plot 1, then tap zoom-9 to see
  the linearized data
• It should look just like your manual graph
• Now perform linear regression of log y against x
  and paste the LSRL into Y2
• In other words, LinReg(abx)L1,L3,Y2
• Note the very high r value of .999
• Tap GRAPH to see the fit
• Turn off the main graph and turn on the residual
  plot
• Sketch the residual plot
• Comment on what the plot says about goodness of
  fit
• Write the equation of the LSRL (round to .001)

8
Converting from LSRL to Exponential Model

• The equation we found says log y ABx
• Raise bases of 10 to both sides and simplify
• 10log y 10(ABx)
• y 10A 10Bx (10A)(10B)x
• Now recall that A log a, so 10 A 10 log a a
• Similarly, 10 B b
• So the equation becomes y abx
• In other words, use LinReg on the linearized data
  to find A and B, then convert to a and b in the
  model we seek.
• The magic formulas are
• To find a, evaluate 10A (where A is the a given
  by LinReg)
• To find b, evaluate 10B (where B is the b given
  by LinReg)

9
Finishing the Starter

• We previously found A and B in the Concord
  population problem
• A 4.002 and B .0211
• So for the exponential model y abx
• a 104.002 10046
• b 10.0211 1.050
• Write the exponential model with the numbers
  filled in
• y 10046(1.050)x
• Put this model in Y3 and graph it with the
  original data (Plot 1)

10
Summary

• Linear regression on the raw data gave a curved
  residual plot, so we tried an exponential model
  instead.
• Put logs of y values in L3 and run LinReg.
• Check linearized data and resid plot.
• Calculator gives A and B, not a and b.
• Convert to a and b enter y abx and graph on
  plot of original data.

11
Objectives

• Convert exponential data to linear data by use of
  logarithm principles
• Perform linear regression on linearized data
• Evaluate linear fit by using a residual plot
• Convert linear results to an exponential function
  of the form y abx that models the original data

12
Homework

• Read pages 176 188
• Do problem 1