Which of the following statements are true when $\bar { X }$ is the mean of a random sample of size n from a normal distribution with mean $\mu$ ?
A) The random variable $Z = ( \bar { X } - \mu ) / ( S / \sqrt { n ) }$
Has approximately a standard normal distribution for large n.
B) The random variable $T = ( \bar { X } - \mu ) / ( S / \sqrt { n ) }$
Has a t-distribution with n-1 degrees of freedom for small n.
C) The normal distribution is governed by two parameters, the mean $\mu$
And the standard deviation $\sigma$
)
D) A t-distribution is governed by only one parameter, called the number of degrees of freedom.
E) All of the above answers are true.

Question

Quizplus · Accepted Answer

A) is true because for large sample sizes, the Central Limit Theorem ensures that the sampling distribution of the sample mean $\bar{X}$ is approximately normal, and thus $Z$ has approximately a standard normal distribution. B) is true because when the population standard deviation is unknown and is estimated by the sample standard deviation $S$, the distribution of $T$ follows a t-distribution with $n-1$ degrees of freedom, especially relevant for small sample sizes. C) is correct as the normal distribution is indeed characterized by its mean $\mu$ and standard deviation $\sigma$. D) is also true because the shape of a t-distribution is determined by its degrees of freedom.

Which of the Following Statements Are True When Xˉ\bar { X }Xˉ

Which of the Following Statements Are True When $\bar { X }$