Let $x$ represent the number which shows up when a balanced die is rolled. Then $x$ is a random variable with a uniform distribution. Let $\bar { x }$ denote the mean of the numbers obtained when the die is rolled 3 times. Which of the following statements concerning the sampling distribution of the mean, $\bar { x }$, is true?
A)$\bar { x }$ is approximately normally distributed.
B) $\bar { x }$ is normally distributed.
C) $\bar { x }$ has a uniform distribution.
D) None of the above statements is true.

Question

Let $x$ represent the number which shows up when a balanced die is rolled. Then $x$ is a random variable with a uniform distribution. Let $\bar { x }$ denote the mean of the numbers obtained when the die is rolled 3 times. Which of the following statements concerning the sampling distribution of the mean, $\bar { x }$, is true?  
A)$\bar { x }$ is approximately normally distributed.
B) $\bar { x }$ is normally distributed.
C) $\bar { x }$ has a uniform distribution.
D) None of the above statements is true.

Quizplus · Accepted Answer

The sampling distribution of the mean, $\bar{x}$, when rolling a die 3 times does not become normally distributed (due to the small sample size and the original distribution being uniform, not normal), nor does it remain uniformly distributed. The Central Limit Theorem, which suggests that the sampling distribution of the mean approaches a normal distribution, typically requires a larger sample size to apply.

Let xxx Represent the Number Which Shows Up When a Balanced Die

Let $x$ Represent the Number Which Shows Up When a Balanced Die