Identify the distribution of the sample mean. In particular, state whether the distribution of $\bar { x }$ is normal or approximatelynormal and give its mean and standard deviation.
-Let $x$ represent the number that shows up when a balanced die is rolled. Then $\mathrm { x }$ is a random variable with a mean of $3.5$ and a standard deviation of 1.71. Let $\bar { x }$ denote the mean of the numbers obtained when the die is rolled 40 times. Determine the sampling distribution of $\bar { x }$.
A)Normal, mean $= 3.5$, standard deviation $= 0.04$
B) Normal, mean $= 3.5$, standard deviation $= 0.27$
C) Approximately normal, mean $= 3.5$, standard deviation $= 0.27$
D) Approximately normal, mean $= 3.5$, standard deviation $= 1.71$

Question

Identify the distribution of the sample mean. In particular, state whether the distribution of $\bar { x }$ is normal or approximatelynormal and give its mean and standard deviation.
-Let $x$ represent the number that shows up when a balanced die is rolled. Then $\mathrm { x }$ is a random variable with a mean of $3.5$ and a standard deviation of 1.71. Let $\bar { x }$ denote the mean of the numbers obtained when the die is rolled 40 times. Determine the sampling distribution of $\bar { x }$.  
A)Normal, mean $= 3.5$, standard deviation $= 0.04$
B) Normal, mean $= 3.5$, standard deviation $= 0.27$
C) Approximately normal, mean $= 3.5$, standard deviation $= 0.27$
D) Approximately normal, mean $= 3.5$, standard deviation $= 1.71$

Quizplus · Accepted Answer

The Central Limit Theorem states that the distribution of the sample mean will be approximately normal if the sample size is large enough (usually $n \geq 30$). Here, $n = 40$, which is sufficient. The mean of the sampling distribution is the same as the population mean, $3.5$. The standard deviation of the sampling distribution (standard error) is the population standard deviation divided by the square root of the sample size: $1.71/\sqrt{40} \approx 0.27$.

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

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