Twelve-sided Dice and the Central Limit Theorem. Rolling a fair 12-sided die produces a uniformly distributed set of numbers between 1 and 12 with a mean of 6.5 and a standard deviation of 3.452. Assume that n 12-sided dice are rolled many times and the mean of the n outcomes is computed each time.

a. Find the mean and standard deviation of the resulting distribution of sample means for n = 81.

b. Find the mean and standard deviation of the resulting distribution of sample means for n = 100.

c. Why is the standard deviation in part a different from the standard deviation in part b?