Section 1'6 Fitting Linear Functions to Data

1
Section 1.6Fitting Linear Functions to Data
2

• Consider the set of points (3,1), (4,3), (6,6),
  (8,12)
• Plot these points on a graph
• This is called a scatterplot
• Sketch a straight line that best fits the given
  data

3

• There is a standard way of picking a line of
  best fit
• Creating a line of best fit is called linear
  regression
• Lets see how to do this in our calculators

4

• The following table gives the closing value of
  the Dow-Jones average, D, for several different
  years, t
• Use your graphing calculators to find the
  equation of the regression line
• Based on the equation predict the Dow-Jones value
  in 1984 and 1987
• Based on the equation predict the Dow-Jones value
  in 1993 and 2000

5
Dow Jones Average
6
Interpolation vs. Extrapolation

• When you estimate the output value for an input
  that is within your extreme values it is called
  interpolation
• When we found the values for 1984 and 1987 we
  used interpolation
• This is considered more reliable because we are
  within an interval we know something about
• When you estimate the output value for an input
  that is outside your extreme values it is called
  extrapolation
• We did this when we found the values for 1993 and
  2000
• This is considered less reliable because we are
  outside the known inter al

7
How regression works

• We assume the value y is related to the value of
  x
• The line is chosen to minimize the sum of the
  squares of the vertical distances between the
  data points and the line
• Such a line is called a least-squares line

8
Correlation

• Way of measuring goodness of fit
• Values between -1 and 1
• Values near -1 or 1 imply strong linear
  relationship between variables
• Values near 0 imply no (or weak) linear
  relationship between variables
• This does not mean there is no relationship
• See figure 1.56 on page 45
• This does not mean one variable causes the other
• Example there is a strong positive correlation
  between boat sales and car sales but one does not
  cause the other