13. On each die in a set of six dice the pips on each face are diamonds. It is suggested that the weight of the stones will cause the "five" or "six" faces to fall downward, and hence the "one" and "two" faces to fall upward, more frequently than they would with fair dice. To test this conjecture, a trial is conducted as follows: The throwing of a one or two is called a success. The six dice are thrown together 64 times and the frequencies of throws with 0,1,2,..., 6 successes summed over all six pairs are as follows: Successes out of six (number of dice 012 3 4 5 6 showing a one or two) Frequency 04 19 15 17 7 2

(a) What would be the theoretical probability of success and the mean and variance of the above frequency distribution if all the dice were fair?

(b) What is the empirical probability of success calculated from the data and what is the sample average and variance?

(c) Test the hypothesis that the mean and variance have their theoretical values.

(d) Calculate the expected frequencies in the seven "cells" of the table on the assumption that the probability of a success is exactly 3.

(e) Can you think of a better design for this trial?