Marine Pollution
Part 3: Fitting
a Line to the Data
Suppose we hypothesize that
a linear relationship exists between the concentration of TBT in the mussels
and the thickness of their shells. As a rough test of this hypothesis we
can try to fit a "good linear approximation" to the data, i.e.,
a linear function whose graph seems to reflect the trend represented by
the data points. A general linear function has the equation
y = ax + b,
where a is the slope
of the line and b is the y-intercept.
- In your worksheet, a command
for defining and plotting y is provided with "dummy" values
of a and b. Experiment with various values of a and
b to make sure you know what the command does.
- Now graph the mussel data
and a "dummy" line on the same plot. Specific instructions are
given in the worksheet.
- Does the line give a good
approximation to the data? If not, then change the values for a
and b, and activate the statement again. Keep trying values of a
and b until you feel you have obtained a good linear approximation
to the data.
- Your computer algebra system
has a command to produce a "best linear fit" to a set of data.*
This command may be associated with the name "least squares fit"
or "regression line." Plot the "best fit" along with
the actual data and your "good fit." (Again, specific instructions
are given in the worksheet.) How close is your choice to the presumed "best"
choice?
- Interpret your choices
for the numbers a and b -- whether you agree with the "best"
choices or not. What does slope a mean in the context of mussels?
What does y-intercept b mean?
*The meaning of "best"
in this context will be explored in later modules on least-squares fitting that
may be found in the multivariable calculus and linear algebra collections.
