2 Datapoints and a (1Parameter) Model:

Context 
y x   2 1 8 2 
One has 2 indep'nt observations from the 'nointercept' model E[yx] = B x. The y's might represent the total numbers of typographical errors on x randomly sample pages of a large document, and the data might be y = 2 errors in total in a sample of x = 1 page, and y = 8 errors in total in a separate sample of x = 2 pages. The B in the model represents the mean number of errors per page of the document. Or, the y's might represent the total weight of x randomly sample pages of a document, and the data might be y = 2 units of weight in total for a sample of x = 1 page, and y = 8 units for a separate sample of x = 2 pages. The B in the model represents the mean weight per page of the document. We gave this 'estimation of B' problem to several statisticians and epidemiologists, and to several grade 6 students, and they gave us a variety of estimates, such as Bhat = 3.6/page, 3.33/page, and 3.45! HOW CAN THIS BE? 
applet 
Click on different (y,x) locations to obtain different fitted values
given by different B's (different y/x ratios),
and keep track of the various 'fit' criteria. Left: Vertical lines (red): the residuals measured on the yscale; Blue lines: the Poisson probabilities of obtaining these 2 y values. Right: red=trace of sum of squares blue=trace of logL Updated: Aug 20, 2012 