Identify the distribution of the sample mean. In particular, state whether the distribution of $\bar { x }$ is normal or approximately normal and give its mean and standard deviation.
-For the population of one town, the number of siblings, $x$, is a random variable whose relative frequency histogram has a reverse J-shape. The mean number of siblings is $1.1$ and the standard deviation is 1.3. Let $\bar { x }$ denote the mean number of siblings for a random sample of size 39 .
Determine the sampling distribution of the mean for samples of size 39 .
A)Normal, mean $= 1.1$, standard deviation $= 1.3$
B) Approximately normal, mean $= 1.1$, standard deviation $= 0.21$
C) Normal, mean $= 1.1$, standard deviation $= 0.21$
D) Approximately normal, mean $= 1.1$, standard deviation $= 1.3$

Question

Identify the distribution of the sample mean. In particular, state whether the distribution of $\bar { x }$ is normal or approximately normal and give its mean and standard deviation.
-For the population of one town, the number of siblings, $x$, is a random variable whose relative frequency histogram has a reverse J-shape. The mean number of siblings is $1.1$ and the standard deviation is 1.3. Let $\bar { x }$ denote the mean number of siblings for a random sample of size 39 .
Determine the sampling distribution of the mean for samples of size 39 .  
A)Normal, mean $= 1.1$, standard deviation $= 1.3$
B) Approximately normal, mean $= 1.1$, standard deviation $= 0.21$
C) Normal, mean $= 1.1$, standard deviation $= 0.21$
D) Approximately normal, mean $= 1.1$, standard deviation $= 1.3$

Quizplus · Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normal regardless of the shape of the population distribution, given a sufficiently large sample size (usually n ≥ 30). The mean of the sampling distribution ($\bar{x}$) is equal to the population mean ($\mu$), which is 1.1. The standard deviation of the sampling distribution ($\sigma_{\bar{x}}$) is equal to the population standard deviation ($\sigma$) divided by the square root of the sample size (n), which is $1.3 / \sqrt{39}$ ≈ 0.21.

Identify the Distribution of the Sample Mean xˉ\bar { x }xˉ

Identify the Distribution of the Sample Mean $\bar { x }$