949
edits
Changes
→In Search of the Truth: Who's Better?
Of the 508 games of cribbage played between Fall 2001 and [[Commencement]] 2005, Zach was victorious in 264 games, Jonathan in 244. However, only 31 of Zach's victories were skunks, to Jonathan's 36. This accounts for the smaller margin of difference in the final score: Zach 295, Jonathan 280.
As [[Psychology]] majors, Jonathan and Zach were eventually enriched by endowed with a knowledge of basic statistical tests, and from the point of learning the chi-squared analysis -- the statistical test with the power to determine, statistically, which of the two was , statistically, a better player-- frequent such tests would be used to analyze the running tally of wins and losses, usually by the player who was ahead.
Interpreting data by a Pearson's chi-square analysis is done using the formula:
:<math> \chi^2 = \sum_{i=1}^{n} {(O_i - E_i)^2 \over E_i}</math>
where <math>O_i</math> is an observed frequency and <math>E_i</math> is an expected (theoretical) frequency asserted by the null hypothesis.
In the case of the 508 games played in the four years of cribbage competition, our expected number of wins for each player is 254. Note that we use the word "expected" here in the cold, impartial sense of the statistician, who generates his expectations on pure probability. In this expectation, therefore, we ignore such obvious cues as, for example, the blondness of one player, the increased height of one player (resulting in a lower brain to total mass ratio), city of birth, etc. We can plug this expected value and the observed number of wins into the equation to determine the <math>\chi^2</math> value for the analysis:
:<math>\chi^2 = {(264_z - 254)^2 \over 254}+{(244_j - 254)^2 \over 254}</math>
We now compare this value of <math>\chi^2</math> to a threshold value of <math>\chi^2</math> in our chi-squared distribution of two degress of freedom. In experiments of this kind, it is customary to take a confidence level of at least 95% as evidence that your data demonstrates a significant trend, however to have a confidence of 95% that one player were better than the other in this case we would need a <math>\chi^2</math> value of '''3.84''' or greater. 0.787... is far too low, corresponding more closely to a confidence of about 40%.
Though it went against the standards of every academic field, including his own basic training in [[Psychology]] Psyc statistics, Zach seemed satisfied by this level of confidence. He wrote, ''There was a time senior year when a t-squared analysis ''[sic.]'' on the number of victories each showed that Zach was, statistically, a superior cribbage player to Jonathan. Once that point was reached, Zach felt his point was proven and lowered his level of play, allowing Jonathan to win a few games and regain a bit of dignity... at least statistically speaking.''
==Cribbage moments==