Line of Best Fit

We frequently observe data involving two variables and are interested in finding an equation relating the two variables. In this section we will look data that suggests a linear equation.

Example

Listed below are the world record times for the mile run and the year in which they were set. a.) Find the line that best fits this data.

b.) Use the equation obtained to predict when the record will be 3 minutes 42.6 seconds.

1967 3 min 51.1 secs or 3.852 min

1975 3 min 49.4 secs or 3.823 min

1979 3 min 49.0 secs or 3.817 min

1980 3 min 48.8 secs or 3.813 min

1981 3 min 47.33 secs or 3.789 min

1985 3 min 46.32 secs or 3.772 min

1993 3 min 44.39 secs or 3.740 min

Solution .

Note that the times have all been converted to minutes so that we can use them in our calculations. In addition we will use 1967 as our base year (i.e. it will correspond to year 0, 1975 will correspond to 8, etc)

We first enter the list of years with 1967 corresponding to 0.

Here is the list of times in decimal form.

We now find the best fitting line predicting the mile time.

Here is a graph of the data and the best fitting line.

To use the model to predict when the record for the mile will be 3:42.6, convert 42.6 seconds to .71 minutes and solve predMileTime=3.71 for x.

Solving -.004458974359x + 3.85882381 = 3.71 for x yields

x = 33.37624261.

Since x is years beyond 1967, the model predicts a time of 3:42.6 some time between the year 2000 and 2001.

Note :

Many calculators have a linear regression (the technique used in this section) feature built into them. The work in this example was done with MenuMaple accessing the statistical features of Maple .