For the population of one town, the number of siblings is a random variable whose relative frequency histogram has a reverse J-shape. Let $\bar { x }$ denote the mean number of siblings for a random sample of size 30. For samples of size 30, which of the following statements concerning the sampling distribution of the mean is true?
A)The distribution of $\bar { x }$ has a reverse J-shape.
B) $\bar { x }$ is approximately normally distributed.
C) $\bar { x }$ is normally distributed.
D) None of the above statements is true.

Question

For the population of one town, the number of siblings is a random variable whose relative frequency histogram has a reverse J-shape. Let $\bar { x }$ denote the mean number of siblings for a random sample of size 30. For samples of size 30, which of the following statements concerning the sampling distribution of the mean is true?  
A)The distribution of $\bar { x }$ has a reverse J-shape.
B) $\bar { x }$ is approximately normally distributed.
C) $\bar { x }$ is normally distributed.
D) None of the above statements is true.

Quizplus · Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution, provided the sample size is sufficiently large (usually n ≥ 30 is considered sufficient).

For the Population of One Town, the Number of Siblings xˉ\bar { x }xˉ

For the Population of One Town, the Number of Siblings $\bar { x }$