TechSkills 2: Regression
In this module we exploit the TI-89's statistical features to obtain a function that best describes a given data set from some point
of view. The user must first decide on a kind of function to use, then the TI-89 finds an equation of best fit for that function.
Once we have a function, then we can apply mathematical techniques for analysis.
Linear regression is the process of fitting a straight line to a data set. A linear function has the form y = ax + b. Hence, the task of linear regression is to determine values for a and b that create the straight line that best fits the given data. For example, we consider a data set from Sullivan and Sullivan, Precalculus enhanced with graphing utilities, Second Edition, page 112, Example 3.
|fertilizer, X (lbs/100ft2)||0||0||5||5||10||10||15||15||20||20||25||25|
|Yield, Y (bushels)||4||6||10||7||12||10||15||17||18||21||23||22|
First we enter the data points into a table in the TI-89. Then we conduct a linear regression to obtain the equation of the straight line of best fit through the three points. The slope of the regression line gives us a best estimate for the average rate of change of yield per pounds of fertilizer applied per 100 square feet.
Step 1: Data Entry
Note If you have used the Editor before, it may be more convenient simply to choose the 1: Current… option. Either way, you will end up in the Editor screen. The main difference is that by going through the 3: New option you get an empty data table. With the 1: Current option, the data table may have entries in it that need to be cleared. Clear the Editor by pressing and then to select 8Clear Editor.
Note If you want, you may name the columns. Use the thumb pad to highlight the cell above c1, then type . Highlight the cell above c2, then type .
After data entry, the next step is to plot the data set and then determine what type of function is appropriate for describing the data.
Step 2: Plotting the Data Set
To plot the data, you must tell the TI-89 where the data are that you would like to plot and what kind of plot that you want.
Note If some unwanted graphs appear on the data set, go to the Y= menu and either deselect (use the key) or clear the unwanted functions.
Now look at the data set. Ask yourself, is it reasonable to describe this data set with a linear function? The linear function need not be a perfect fit to the data set, but should act as a general descriptor of the trend of the data in the viewing window. Here we notice the general upward trend of the data. It appears reasonable to draw a straight line through the data set that would act as a basic description of the general trend of the data.
Note You can use the Trace feature ( key) to trace the data points on your StatPlot. You may also adjust the tic marks by adjusting the WINDOW menu; for example, set xmin=-1, xmax=26, xscl=5, ymin=0, ymax=25, yscl=5, xres=1. After adjusting the window settings, then GRAPH.
Step 3: The Calculations
Now we fit a straight line through the data points. Of course, a straight line is uniquely determined by two points. Although our twelve points don't fit exactly on any straight line, the TI-89 has a built in feature called linear regression that determines the straight line that best fits our points. In statistics, this line of best fit is called a regression line.
entries. Tell the calculator which columns the x (input variable) and y (output variable) values are in. Here, the input values are in column c1 and the output values are in column c2. To enter c1 in the x-box, type in the x-box. Navigate to the y-box by pressing the down arrow key on the thumbpad (do not press the enter key to navigate between boxes). Then in the y-box.
Step 4: Plotting the Data with the Regression Curve
We now obtain a plot of the data set with the regression line superimposed on the data. Since the plot setup is already done, we merely need to ask for the graph.
Note You can use the Trace feature to trace either the data points or the regression line. Press the key to access Trace. Use the left-right thumb pad keys to move along the data set. Use the up (or down) key to transfer to the regression line. Then, the left-right thumb pad keys will navigate along the regression line. Use the up (or down) key to return to the data set.
To illustrate obtaining a function from data, we utilize a data set for the growth of the brewers' yeast, Saccharomyces cerevisiae, obtained by the Swedish biologist Tor Carlson in 1913. Here, time is measured in hours and population in biomass units.
First enter the data set. Press , then select either 1:Current or 3:New to return to the spreadsheet. If you select 1:Current, you will need to clear the old data. To clear the old data in column c1, first use the arrow keys to highlight the column heading c1. Then press to access F6 Util, the utilities menu. Then select 5:Clear Column. Then use the arrow keys to highlight the column heading c2. Then press F6 Util again, and again select 5:Clear Column. After the old data set is cleared, then enter the new data set.
After the data are entered and checked, then we are ready to view the scatter plot. Press , then GRAPH (the display may not show your data in a satisfactory manner). First, go to the Y= menu and clear or deselect any unwanted functions. Then, use the handy ZoomData function by pressing F2 to access the Zoom menu, then press to select 9:ZoomData. Your data set is then nicely displayed in the viewing window. Although the data points are plotted, we can adjust the window for better viewing. First go to the Y= menu and clear the previous regression equation. Then adjust the WINDOW with settings appropriate for the yeast data; for example: xmin=-1, xmax=8, xscl=1, ymin=0, ymax=300, yscl=100, xres=1. Finally, press , then GRAPH.
Step 3: The Calculations
We are now ready to fit a function to the data. For this we must first decide on a function whose graph looks like a good descriptor for the data. From merely looking at the data and mentally overlaying a smooth curve through the points, we envision a curve that is increasing and describing growth, suggesting an exponential function. No calculator or computer can make this decision. You must make this decision using your knowledge of mathematics and the science of the background reality. For the TI-89, the general exponential function is y=abx. Using exponential regression, the TI-89 will determine the values for a and b that provide a general exponential function of best fit based on the given data.
Step 4: Plotting the Data and Regression Curve
Since the plot setup has already been done in viewing the data set, we are ready to graph.
Note You can use the Trace feature to trace either the data points or the exponential curve on your StatPlot.
One does logistic regression by following the directions given in Nonlinear Regression. However, the equation returned by the TI-89 for logistic regression is not the basic logistic equation. The general form of the basic logistic equation is
where K is the carrying capacity and r is the intrinsic growth rate. However, the TI-89 returns the logistic with a vertical translation in the form
Accordingly, the initial population, f (0), and carrying capacity, K, are calculated by
and K = .
We conclude with an additional step for students in calculus.
A Tangent Line
Your TI-89 can draw a tangent line to a function at a specified point. The slope of the tangent line is our desired estimate for the instantaneous rate of growth; say, at t=4. Here is how it works. We suppose that the screen is showing a plot of the data and a graph of the exponential regression curve.
|Soon the tangent line will be drawn and its equation displayed at the bottom of the screen. Did you get a slope of 32.5? We conclude, the yeast population is increasing at the rate of about 32 biomass units per hour after four hours of growth. Recall that the conclusion to any applied problem should always be stated as a complete sentence with numbers described within the context of the given problem.|
Suppose now that you would like to estimate the slope of the tangent line at another input value. If you want to leave the old tangent line for comparison, proceed with the above instructions for obtaining the tangent line at a new input value. If you'd like to clear the current tangent line, then select F6, then press to select the 1:ClrDraw option to clear the drawing. Enjoy!
The author wishes to extend his appreciation to Texas Instruments for their professor assistance program. Visit the TI calculator website at http://www.ti.com/.
Charles M. Biles, Ph.D.
Department of Mathematics
Humboldt State University
Arcata, CA 95521-8299