Fitting Formulas to Data: TI86

The TI86 offers the following options for curve fitting. These can all be found by pressing [STAT] (which is [2nd][+]), and then by selecting option CALC (which is the [F1] key).

Description	Command	Formula
Linear regression	`LinR`	y = a + b x
Logarithmic regression	`LnR`	y = a + b ln(x)
Exponential regression	`ExpR`	y = a b^x
Power regression	`PwrR`	y = a x^b
Sinusoidal regression	`SinR`	y = a sin(b x + c) + d
Logistic regression	`LgstR`	y = a (1 + b e^{c x})^-1 + d
Quadratic regression	`P2Reg`	y = a x² + b x + c
Cubic regression	`P3Reg`	y = a x³ + b x² + c x + d
Quartic regression	`P4Reg`	y = a x⁴ + b x³ + c x² + d x + e

The steps for fitting a regression curve to a collection of data points are as follows:
1. Enter the data
2. View a scatter plot
3. Obtain an appropriate regression equation
4. Plot the regression equation along with the scatter plot
Example 1: Our first example will be represented by the following table:

a 3 4 5 6

S 105 117 141 152

We wish to find a curve that best fits this set of points.
1. Enter the data
  
  The calculator stores data in lists. We will use the lists specified xStat and yStat. The steps are:
  1. Clear the lists:
    
    Use [STAT] with option EDIT (press the [F2] key) to get the lists on the screen. If xStat and/or yStat have data already, then each list can be cleared by moving the cursor up to the name of the list, at the top, and then by pressing [CLEAR] followed by [ENTER].
  2. Move the cursor into the xStat column. From the bottom of the screen, below the lists, enter each data item from the line of the table that corresponds to the values of a, followed by [ENTER].
    
    Next move the cursor into the yStat column and repeat for each element in the line of the table that corresponds to the values of S.
    
    Next move the cursor into the fStat column. For each element of data in the xStat and yStat columns you must input the value 1 in the fStat column. Note that this is needed in order to obtain the regression equation, but not in order to obtain a scatter plot of the data.
    
    When the data have been entered, [QUIT] the screen (which is [2nd][EXIT]).
2. View a scatter plot:
  1. Any undesired functions entered on the \y1=, \y2=, ... lines of option y(x)= of [GRAPH] will have to be cleared out, regardless of whether Plot1 is On or Off. To do this, go to [GRAPH], select option y(x)= (which is [F1]), move the cursor to each appropriate line and press [CLEAR]. Exit by pressing [QUIT].
  2. To turn Plot1 On or Off, go to [GRAPH], and select option y(x)=. If Plot1 is encased in a dark rectangle, then it is On. It is Off otherwise. To change the On/Off status of Plot1, move the cursor to the top of the screen, to Plot1, and toggle it by pressing [ENTER]. Once the proper status is designated, press [QUIT].
  3. To change the type of points that Plot1 is designated to produce, go to [STAT], select option PLOT (which is [F3]), and then select option Plot1 (which is [F1]). Move the cursor down to Mark=, and select the point option desired--either a small square [F1], a plus sign [F2], or a dot [F3]. When the cursor is in position, press [ENTER]. Once this is selected, press [QUIT].
  4. There are a number of methods of generating a scatter plot. We consider two principal methods here.
    1. The first method is based on the functions found in [GRAPH]. It only works if Plot1 is On.
      
      Make sure that Plot1 is On. If so, then go to [GRAPH], select option ZOOM (press the [F2] key), and then select option ZDATA (which consists of pressing [MORE] and then the [F5] key). A scatter plot of the data will then appear. The points will appear as whatever is designated by Plot1.
      
      The x- and y-dimensions of the screen window will be set automatically. Look at the window settings (option WIND, which is [F2] under [GRAPH]) to see which parameters were chosen.
    2. The second method has two variations and is based on the functions found in [STAT]. The variations depend on whether Plot1 is On or Off, and so the status of Plot1 should be determined beforehand.
      
      This method is less favorable because the x- and y-dimensions of the screen window are not set automatically, and may need to be altered manually to get a good picture of the data (option WIND under [GRAPH]).
      1. If Plot1 is Off, then go to [STAT] and select option DRAW (press the [F4] key). From there, select the additional option SCAT (press the [F2] key) of DRAW. A scatter plot of the data will then appear. The points will appear as small dots on the screen, and are a little hard to see.
      2. If Plot1 is On, then go to [STAT] and select option DRAW. A scatter plot of the data will appear immediately after this selection. The points will appear as whatever is designated by Plot1.
        
        As we already have a scatter plot of the data, we do not need to select the additional option SCAT of DRAW. However, if it is selected, then small dots will appear in addition to what is already on the screen. This becomes noticeable only if the point option of Plot1 is set to the small square configuration.
  5. To clear the option bar at the bottom of the graphics screen, press [CLEAR], and to bring it back, press [STAT].
3. Obtain an appropriate regression equation:
  1. The scatter plot suggests that the relationship between measurements a and S is approximately linear, so we will choose a linear regression.
  2. Go to [STAT] and choose option CALC; then choose option LinR (which is the [F3] key). On the screen, LinR will appear. At this point, press [ENTER].
    
    This yields the values a=54.5 and b=16.5 for the equation y=a+bx, so that our regression equation is
    
    y = 16.5 x + 54.5.
    The correlation coefficient, corr=.988626763, indicates how well the regression curve fits the data. A correlation that is close to 1 is good.
4. Plot the regression equation along with the scatter plot:
  
  Under [STAT], select option DRAW (which is [F4]). Then select option DRREG (which is [MORE] and [F1]), and the graph appears on the screen, along with the scatter plot.
Example 2: Here we use the data from Problem 4 on page 5, which gives the profit (in millions of dollars) for The Gap, Inc. for the years from 1990 to 1997.

Year (since 1990) 0 1 2 3 4 5 6

Sales ($ million) 144.5 229.9 210.7 258.4 350.2 354.0 452.9

Once again, we wish to find a curve that best represents the relationship between the two variables.
1. Enter the data as we did earlier.
2. View the corresponding scatter plot.
  
  The scatter plot looks fairly linear, but, since this is financial data, we want a growth rate, so we will look for an exponential model.
3. Follow the same steps to obtain the regression equation, but this time choose ExpR (which is [F5]), under the CALC option in [STAT].
  
  This yields the values a=160.783898 and b=1.18693787 for the equation y=a*b^x, so that our regression equation is approximately
  
  y = 160.8 (1.187)^x.
  In this example, the correlation coefficient is corr=.962051541.
4. Recall that our model is
  
  P = P₀ (1 + r)^t.
  We have found the growth factor 1 + r to be approximately 1.187. This means that the growth rate r = 0.187. We can conclude that the profits for The Gap, Inc. were growing at 18.7% for this time period.
5. Follow the same steps that we had with the previous problem to view the exponential graph with the scatter plot.

Last modified: Tue Sep 11 2001

Year (since 1990)	0	1	2	3	4	5	6
Sales ($ million)	144.5	229.9	210.7	258.4	350.2	354.0	452.9

a	3	4	5	6
S	105	117	141	152