## Edible and Tacky Experiments to Model Exponential Decay

The Classic M-&-M Spill Experiment
• Divide the class into groups of four; give each group two paper cups, one of which contains exactly 100 M-&-M candies. Each group should fold a sheet of paper to form a tray with a lip about one inch high.

• Each group is to shake their cup and spill their M-&-M's into their tray. Count the the number of M-&-Ms that land "M"-side-up, then remove them from the tray and place them in a "holding" cup. In a table with three columns labelled "# Spills", "# M-side-up", and "# Remaining", record the appropriate numbers. Put the remaining M & Ms back into the shaking cup.

• Repeat this cycle of (spilling the candies into the tray) > (counting and removing those that land M-side-up) > (recording the number removed) until there are no candies left.

• Enter the recorded data into lists on the TI-83, using [STAT] >> EDIT.
Enter the number of Spills into L1, the number that landed M-side-up into L2, and the number Remaining into L3.
Do NOT enter the data for the last spill that results in 0 candies left. If you do, you will get a "Domain" error message when trying to get the regression equation. Can you explain why that is so?

Use [2nd]-[STATPLOT] to set Plot1 up as a scatterplot of the data, with L1 for the Xlist, and L3 for the Ylist, since we are first going to study how the number remaining decreases; select the square as the mark. Set up the [WINDOW] with appropriate values for Xmin, Xmax, Ymin and Ymax. Graph the scatterplot representing the data in your lists.

• We would like to find a mathematical model that fits this data. Using [STAT] >> CALC shows that the calculator can calculate parameters to "force" the data into such mathematical models as linear, quadratic, exponential, power, etc. Let's begin by finding a linear function to model your data. Let the calculator simultaneously calculate the regression equation and store it as Y1 by using the sequence of keystrokes:
[STAT] >> CALC >> 4:LinReg >> [2nd]-[L1] >> [,] >> [2nd]-[L3] >> [,] >> [VARS] >> Y-VARS >> 1:Function >> [ENTER] >> [specify Y1] >> [ENTER] >> [ENTER].

• We can now look at both the scatterplot of the data and the calculated regression model, and think about how well the function represents the data.
When the regression equation is calculated, the machine also diagnoses how well the particlar regression equation matches the data, displayed as "r2 =" and "r =". If these did show automatically, you use [2nd]-[CATALOG] >> Diagnostics On to turn this functionality on.

• NOTE: For a more advanced approach, we can get a better sense of how closely the data fits the function, the TI-83 can show the distance between each data point and the regression function evaluated at each X value. These distances are described as "residuals", and are automatically evaluated. To place the residual values in a new list column:
[STAT] >> EDIT >> (move the cursor onto the label of L4) >> [2nd]-[INS] (which gives a prompt for a name) >> [2nd]-[LIST] >> (this gives a list of names; arrow down and select RESID) >> [ENTER] >> [ENTER].
The label "RESID 1" should appear at the top of the column next to L3, but the values will not be there. To show the values you must re-calculate the regression equation; the residuals will appear automatically in the RESID column.
The importance of residuals is that they can confirm that the model is close, or that the model is off in a random way, or that there is a pattern in how the model does NOT fit the data.
To see a graphical representation of the residuals, we will set up a second scatterplot:
[2nd]-[STATPLOT] >> 2:Plot2 >> [ENTER]; set Xlist as L1, and Ylist as RESID; use the "+" mark so that they are distinguishable from data points. In the Y= window, turn both Y1 and Plot 1 off, then turn Plot 2 on.

Is there a discernable pattern in the residuals in this situation? If so, what does it tell you?

• It should be clear from the graph as well as from the lists, that this is not a great fit. This suggests that a different type of function might model the data better.
Let's try an exponential function, with a form of Y = a*b^X, which typically decreases like the data (as long as the the b < 1).
In order for the residuals to be clearly related to our new regression function, [CLEAR] the linear equation from Y1.

• To calculate an exponential equation from the data, use the sequence of keystrokes:
[STAT] >> CALC >> 0:ExpReg >> [2nd]-[L1] >> [,] >> [2nd]-[L3] >> [,] >> [VARS] >> Y-VARS >> 1:Function >> [ENTER] >> [ENTER] >> [ENTER].
Showing both Y1 and Plot 1 will allow you to see the data points along with the exponential regression function.
You can examine the residuals as a list, or by showing Plot 2, to help you analyze how well this model fits the data.
Questions
1. Does an exponential decay model, in fact, offer a better fit with the data?

2. The exponential function is of the form Y = a*b^X.

• What meaning, if any, do the values of the parameters a and b have with regard to this experiment?

• What could we reasonably expect for values for a and b in this situation? How do you know this?

• We might get better results if we include the initial number of M & Ms. Enter this data as Spill # 0, with 0 M-side-ups, and the initial value as the # remaining. Recalculate the exponential regression equation and paste it into Y2. Does this second effort get a better fit?

The Neo-Classic Thumbtack Spill Experiment
• We will replicate the process from the M-&-M Spill Experiment, but substitute thumbtacks for M-&-Ms.

• What differences do you anticipate this might make in the regression equation, and why?