A B C D

Notes for Session 8, Part D

 Note 9 In Part D, we return to the statistics question in Part A, based on the game of Push Penny: "After several practices of Push Penny, have you developed skill in playing the game?" One approach is to compare the data (the 100 scores of a player) to the expected scores of a "random" player (one with no particular skill, who is making "random" pushes). This strategy requires a model for a "random" player, which must be based on probabilities, because there is randomness in the outcomes of the games. A game consists of four pushes. First, you consider the probability that a single random push will hit a line. Experiment with a quarter on the Push Penny board to investigate this. The key is to discover that the lines are uniformly spaced (the distance between lines is equal to two times the diameter of a quarter). By moving a quarter perpendicularly to the lines, you'll discover that the coin is touching a line half of the time and not touching a line half of the time.

 Note 10 The leap from playing the game to describing the outcomes with a binomial model can be challenging. To further illustrate how the coin-tossing model describes a random player, the tree diagram is revisited. When the tree diagram for possible scores of this game is discovered to be the same as the tree diagram for the possible number of heads on four coin tosses, the equivalence of the modes may then be more clear.

 Note 11 This analysis is based on investigating how the experimental results compare with the mathematical probabilities. As you'll see, there are very slight differences between the player's scores and the probabilities for a random player. Therefore, there does not appear to be strong evidence that our competitor is any better (or worse) than a random player. This type of analysis is referred to as "goodness of fit" because we are asking how well the model fits the data. A more advanced analysis would require a Chi-Square test, which considers whether the observed differences between the experimental proportions and the theoretical probabilities can be explained by the random variation alone, or if the differences are due to other factors (skill, for instance).