In a set of six very expensive dice the pips on each face are diamonds. It is suggested that the weights of the stones will cause "five" and "six.'' faces to fall downward and hence the· "one" and '' two" faces to fall upward more frequently than they wou1d with technically fair dice. To test the conjecture a trial is conducted as follows: The observance of a one or two on a die is called a success. The whole set of six. dice is then thrown 64 times and the frequencies of throws with 0, 1, 2, . . . , 6 successes are as follows:

(a) What would be the theoretical probability of success and the mean and variance of the 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 vruiance?

(c) Test the hypotheses 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 success is exactly 1/3.

(e) Make a chi-square test of agreement of the observed with the expected frequencies (combining the frequencies for zero and one successes and also for five and six successes so that the expected frequency within each cell is greater than 5). The number of degress of freedom for the test is the number of frequencies compared less 1, since the total of the expected frequencies has been chosen to match the actual total number of trials.

(f) Calculate the· expected frequencies based on the. empirical estimate of probability, and make a chi-square test of-agreement with the observed frequencies. The number of degrees of freedom is now the number of frequencies compared less 2, since expected frequencies have been chosen so that the total frequency and the mean both match the data.

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