MathsNet: Growing dandelions on an EL9600

Growing dandelions on an EL9600

A statistical simulation and analysis, aimed at A level statisticians. Developed from an idea by Jo Tomalin (email) and Dave Cassell.

200 hundred dandelion seeds are scattered at random on a 10 by 8 patch of soil. They all grow. The patch is then subdivided into 80 plots each measuring 1 by 1 and the number of dandelions in each plot is counted. How many plots would you expect to have no dandelions in? How many with just one dandelion? What kind of distribution is involved? How could you model it?

The answer is... find out by growing them on your Sharp Graphical Calculator, model EL9600! Through programming, the calculator can simulate the planting of the seeds, the splitting up into 80 plots and the counting of seeds in each plot. Then the data can be processed into a frequency distribution and finally modelled by either the binomial or poisson probability distributions. From the student's point of view, there are many aspects of this process to appreciate. The programming itself is worth studying, and the various stages of the simulation since they are all key elements in any statisics course. A good way of presenting this simulation to a class is to use the OHP screen that is available for the EL9600.

First a program is required to simulate the planting of the seeds.

ClrG ClrDraw DrawOFF Input T 81->dim(L1) 81->dim(L2) 0->Xmin 10->Xmax 0->Ymin 8->Ymax 0->N Label LOOPN N+1->N N->L1(N) 0->L2(N) If N<80 Goto LOOPN 0->M Label LOOPM M+1->M (10random)->A random->B (8random)->B PntOn(A,B) (10*(ipart B)+(ipart A)+1)->N L2(N)+1->L2(N) If M< T Goto LOOPM Wait 0->N Label LOOPP N+1->N Line(0,N,10,N) If N<8 Goto LOOPP 0->N Label LOOPQ N+1->N Line(N,0,N,8) If N<10 Goto LOOPQ Wait	To run the program, first enter PlotsOff, then enter the PRGM screen and execute this program. It will prompt you for a value for T, say 200, which represents the number of dandelions to be planted. The program clears all lists and displays and sets up the axes. It sets up list L1 to contain the plot numbers 1 to 80. A loop is then run that creates and plots random points (the dandelions) in the 10 by 8 grid and keeps a tally of how many points there are in each of the 80 plots. (The EL9600 seems to have a kind of bug with its random number generator here, which is why there is that extra random->B line. See Some oddities.) When all the points have been displayed, the grid showing the 80 plots is added.
max(L2)+1->P P->dim(L3) P->dim(L4) 0->N Label LOOPR N+1->N N-1->L3(N) 0->L4(N) If N<P Goto LOOPR 0->N Label LOOPS N+1->N L2(N)+1->Q L4(Q)+1->L4(Q) If N<80 Goto LOOPS max(L4)+5->Ymax max(L3)+1->Xmax DrawON Plt1(Hist,L3,L4) DispG Wait	This section of the program uses lists L3 and L4 to create a frequency count of the data in lists L1 and L2. The data is then displayed as a histogram.
T/80->W dim(L3)->dim(L5) dim(L3)->dim(L6) max(L3)->N W/N->P 80pdfbin(N,P,L3)->L6 L3+.5->L5 Plt2(xyLine.,L5,L6) DispG Wait 80pdfpoi(W,L3)->L6 Plt2(xyLine.,L5,L6) DispG Wait	Finally this section works out the theoretical expected frequencies, based first on the Binomial distribution and secondly the Poisson distribution. The number of dandelions planted (T)divided by the number of plots (80) gives the average number of dandelions per plot, stored as W. For the Binomial distribution, the total number of "successes", N, and the probability of success, P, are found and then used in calculating the binomial probabilities. W is used for the first parameter in calculating the Poisson probabilities. (Note that 0.5 is added to the values in list L3 in order than the points are plotted in the middle of each bar of the histogram.)

The simulation can be run many times for various values of T. The illustrations above show an example of T=400. Higher values should provide a better "fit". If this does not happen, then you can ask why. Either the model is wrong or the data isn't randomly distributed. Here are some more displays comparing the results with the expected Poisson frequencies:

The Poisson distribution is usually found to give a better "fit" than the Binomial, particularly for large values of T, which couuld be measured, of course, using the "Chi-squared" distribution.

The simulation could be developed to include considerations like:

some dandelions dying; a probability of survival could be associated with each seed
a dandelion's death affecting its neighbours; dandelions within say 0.1 units from a dead seed having an increased probability of dying
various types of soil in the different plots; a probability of survival could be assocated with each plot.
inclusion of the "chi-squared" test for goodness of fit.

(Thanks to Prof. Dr. L. Paditz for checking the program listing. At his site you will find a version of this program for the Casio CFX-9850G PLUS.
Jo Tomalin has adapted the program and teaching materials for a Casio 7400 and the TI80 and can be emailed for a copy.)