TI-85
TechSkills 2: Regression

In this module we exploit the TI-85'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-85 finds an equation of best fit for that function. Once we have a function, then we can apply mathematical techniques for analysis.

The process of obtaining a function of best fit is called regression. The TI-85 supports a variety of functions for regression: linear, quadratic, cubic, quartic, power, exponential, and logarithmic. We first illustrate with linear regression.

Linear Regression
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.

Plot 1 2 3 4 5 6 7 8 9 10 11 12
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 set into the calculator. In the TI-85 we enter the input data (here, fertilizer) as xStat and the output data (here, yield) as yStat.

Step 1: Data Entry
1. Press Sta.gif
2. Press f2.gif to select EDIT.
3. If needed, use the arrow keys to position the cursor after xlist Name=. If xlist Name=xStat, press Ent.gif; otherwise, Press f1.gif to select xStat, then press . If ylist Name=yStat, press Ent.gif; otherwise, Press f2.gifto select yStat, then press Ent.gif.
4. If needed, clear any old data by pressing f2.gifto select CLRxy. Enter the data. For each data entry, type in a number, then press Ent.gif. Here, x1 = 0, y1 = 4, x2 = 0, y2 = 6, x3 = 5, y3 = 7, etc.
5. When data entry is complete, press Exi.gif twice to return to the home screen.
ts2fig1.gif

Step 2: Plotting the Data Set

ts2fig2.gif

1. Press Gra.gif.
2. Press f2.gif to select RANGE. Enter an appropriate viewing window. Here, xMin = -1, xMax = 26, xScl = 5, yMin = 0, yMax = 25, yScl = 5 work well. After the range is set, press Exi.gif to return to the home screen.
3. Press Sta.gif. Then press f3.gif to select DRAW.
4. Press f2.gif to select SCAT (scatter plot). After the scatter plot appears, then press Exi.gif.

Note If an extraneous graph appears with the data set, press f1.gif to select y(x)= and clear any functions present (use the arrow keys to highlight the function, then press Cle.gif). Then Exi.gif and reDRAW the graph.

Step 3: The Calculations
Now we fit a straight line through the data points. Of course, two points uniquely determine a straight line. Although our twelve points don't fit exactly on any straight line, the TI-85 has a built in feature called linear regression that determines the straight line that best fits the data. In statistics, this line of best fit is called a regression line.

1. Press f1.gif to select CALC. Press Ent.gif twice to select xlist Name=xStat and ylist Name=yStat.
2. Press f2.gif to select LINR (linear regression).
ts2fig3.gif

The screen displays the results of the linear regression: the regression equation is y = a+bx with y-intercept a = 4.7857 and slope b = .7171. We can report that the average rate of change of crop yield is .7 bushels for each pound of fertilizer applied per 100 square feet.

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.

ts2fig4.gif 1. Press Exi.gif, then press f3.gif to select DRAW.
2. Press f4.gif to select DRREG (draw regression). Press Exi.gif. You now have a nice view of the scatter plot with the regression line.

Nonlinear Regression
table.gif

To illustrate obtaining a function from data, we utilize a data set for the growth of the brewers' yeast, Saccharomyces cerevisiae, obtained by the biologist Tor Carlson in 1913. Here, time is measured in hours and population in biomass units.

First enter the data set. Press Exi.gif to return to the home screen. Then press Sta.gif, then f2.gif to select EDIT. Press Ent.gif to retain xlist Name=xStat, then press Ent.gif again to retain ylist Name=yStat. To clear the old data, press f4.gif to select CLRxy. Now, enter the new data set. After the data are entered and checked, then we are ready to view the scatter plot.

ts2fig4.gif 1. Press Gra.gif.
2. Press f2.gif to select RANGE so we can set an appropriate viewing window for the data. Here, we suggest xMin = -1, xMax = 8, xScl = 1, yMin = -25, yMax = 275, yScl = 25 (these settings give a border to the scatter plot so that the lower menus don't obstruct the view). Then press Exi.gif to return to the home screen.
3. Press Sta.gif, press f3.gif to select DRAW, press f2.gif to select SCAT (scatter plot), then press Exi.gif. You now have a view of the scatter plot.

Step 3: The Calculations
We are now ready to fit a function to the data. First we must decide on a function whose graph looks like a good descriptor for the data. Here, we choose an exponential function. For the TI-85, the general exponential function is y = abx. Using exponential regression, the TI-85 will determine the values for a and b that provide a general exponential function of best fit based on the given data.

The calculations and plots are done exactly as in linear regression, with one exception. In the STAT menu, instead of selecting LINR for linear regression, we select EXPR for exponential regression. For convenience, we summarize the calculations and viewing the scatter plot with the regression equation superimposed on the data set.

1. Press f1.gif to select CALC.
2. Press Ent.gif to select xlist Name=xStat.
3. Press Ent.gif again to and ylist Name=yStat.
4. Press f4.gif to select EXPR, exponential regression.
ts2fig5.gif

The screen now displays the results of the exponential regression: the regression equation is y = abx where a = 10.9757 and b = 1.5898.

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.

1. Press Exi.gif.
2. Press f3.gif to select DRAW.
3. Press f4.gif to select DRREG. Watch the TI-85 draw the exponential regression line over the scatter plot.
4. When drawing is complete, press Exi.gif to put get a better view.
ts2fig6.gif

We now can forecast based on the regression equation. For example, let's use the regression equation to predict the population at time t = 8. Press f4.gif to select FCST (forecast). Press 8.gif Ent.gif to obtain x = 8. Then press f5.gif to select SOLVE. We predict about 448 biomass units after 8 hours of growth. ts2fig7.gif

END


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
Email: cmb2@axe.humboldt.edu