Section 1.6 Fitting Linear Functions to Data

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

There is a standard way of picking a ‘line of best fit’ Creating a line of best fit is called linear regression Let’s see how to do this in our calculators

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 t D

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 interpolation –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

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

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