Rolling a fair twelve-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 twelve-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-49.
The mean of the resulting distribution of sample means is?
The standard deviation of the distribution of sample means is?

b. Find the mean and standard deviation of the resulting distribution of sample means for n = 121.
The mean of the resulting distribution of sample means is?
The standard deviation of the distribution of sample means is?

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

Given that rolling a fair twelve-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 twelve-sided dice are rolled many times and the mean of the n outcomes is computed each time.

By central limit theorem we can say the mean of all the n outcomes will follow a normal distribution with mean = 6.5 and std deviation = [tex]\frac{3.452}{\sqrt{n} }[/tex] where n stands for number of samples

Thus we find that std deviation of sample mean called standard error is inversely proportional to square root of sample size.

a) Mean = 6.5 and std deviation = 0.4931

b) Mean = 6.5 and std deviation = 0.332

c) The standard deviation is small in part b because here we divide 3.452 by 11, whereas in b we divided by 7 only.