Assume that for an in-control process, the random variable of interest X has a normal distribution with mean value $\mu$ and standard deviation $\sigma$ . If $\bar { X }$ denotes the sample mean for a random sample of size n selected at a particular time, then
A) $E ( \bar { X } ) = \mu$
B) $\sigma _ { \bar { x } } = \sigma / \sqrt { n }$
C) $\bar { X }$
Has a normal distribution.
D) All of the above are true.
E) None of the above are true.

Question

Quizplus · Accepted Answer

Each statement is true regarding the properties of the sample mean $\bar{X}$ in a normally distributed population. A) The expected value of the sample mean is equal to the population mean $\mu$. B) The standard deviation of the sample mean, $\sigma_{\bar{x}}$, is equal to the population standard deviation $\sigma$ divided by the square root of the sample size $n$. C) The sample mean $\bar{X}$ will have a normal distribution due to the Central Limit Theorem, especially as the sample size $n$ becomes large.

Assume That for an In-Control Process, the Random Variable of Interest