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 3 . For samples of size 3 , 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 3 . For samples of size 3 , 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

Given the population distribution has a reverse J-shape and the sample size is only 3, the sampling distribution of the mean $\bar{x}$ would not necessarily be normal or maintain the reverse J-shape of the population. The Central Limit Theorem, which suggests the sampling distribution becomes approximately normal, typically requires larger sample sizes to apply.

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

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