Steve Lekson (1999: 114-123) has argued that Paquime in Chihuahua was intentionally located precisely due south of Chaco Canyon. In fact, using today's best available technology, we know that the alignment is only about 1 km off, over a distance of 630 kilometers. Lekson argued that the Chacoans surveyed from mountain range to mountain range with considerable accuracy to achieve this result. Lekson's critics argue that the alignment is too close-that it could not possibly have been surveyed that accurately with the rather straightforward technology then available, and they suggest that the alignment is coincidental.
Lekson's idea of the surveying process is that you start at Chaco, you figure out where true south is (the best you can) and using mirrors (or whatever) you locate a spot as far as you can see true south of where you are. You then go to that spot and repeat the process until you get to the Rio Casas Grandes. He says he could do it with some boy scouts some mirrors, and some ropes. The beauty of this process, as Lekson points out, is that if the angular errors are random, they have a tendency to cancel each other out. Lets assume that there were 10 survey segments with lengths of {58.4 53.8 95.3 73.8 18.4 24.6 26.1 122.9 21.5 135.2}. Lekson suggests that the Chaco folk were able to shoot alignments within about 2 degrees. Lets assume that means that the angles were randomly distributed between -2 and +2 degrees of true south, based on the alignments of building apparently lined up with celestial phenomena.
Your job is to simulate this process. From a large number of simulated surveys with random measurement errors figure out the mean error after 630km and the probability of getting an error of 1 km or less. That is, if our simulation of the process is realistic, how likely were the Chacoans to get an accuracy as good or better than the one achieved in fact. How would you interpret the results based on what I have told you?
Basically what you want to do is assume that you start at one point and keep track of your east-west errors (you don't actually have to keep track of your north-south progress). Then traverse the 10 segments with the random errors, at each point figuring out your east west error and adding or substracting it froim the cumulative error. I am sure you all remember your trigonometry, but in case you are a little fuzzy here is how I would do it. The east-west error is the length of a given segment (e.g. 58.4km) times the tangent of the angular error.
You need to know a couple of other things. Math.tan(angleRadians) gives you the tangent of the angle where the angle is given in radians. To convert degrees to radians divide by 360 and multiply by 2*pi. This should do it:
angleDegrees=Math.random()*4.0-2.0;
error=Math.tan(2.0*Math.PI*angleDegrees/360.0)*segment;
Notice how I got a random number between -2 and plus 2.
Finally you need to accumulate the results, lets say over several thousand runs. You know how to get the mean. To get the likeli hood of getting a result less than or equal to an observed, simply count the number of times the random result is less than or equal to the observed and then divide by the number of runs.
Lekson, Stephen H. 1999. The Chaco Meridian: Centers of Political Power in the Ancient Southwest. Altamira, Walnut Creek.