Question

Tossing a fair die has outcomes {1,2,3,4,5,6} with mean 3.5 and standard deviation 1.708. If a fair die is thrown three times, and the mean of the resulting triplet is calculated, the mean and standard deviation of the set of all possible such triplets is ____________.

Answer 1

If a fair die is thrown three times, and the mean of the resulting triplet is calculated, the mean and standard deviation of the set of all possible such triplets is 3.5 and 0.986.

Explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

`\mu = 3.5, \sigma = 1.708, n = 3, s = (1.708)/(√(3)) = 0.986`

So the answer is:

If a fair die is thrown three times, and the mean of the resulting triplet is calculated, the mean and standard deviation of the set of all possible such triplets is 3.5 and 0.986.