Question

question #1
which of the following statements are true?
i. the larger the sample, the smaller the standard deviation of the sampling distribution.
ii. sampling distributions of the mean will be centered at the mean of the population.
iii. sampling distributions from non - normal populations are approximately normal provided n is large.

o i, ii, and iii
o iii only
o i and ii only
o ii only

question #2
a sampling distribution of the means of all possible samples of size 100 is formed. the parent population has a mean of μ = 6.7 and a standard deviation of σ = 2.8. what is the mean of the sampling distribution?

o 2.8
o 4.0
o 2.4
o 6.7

Explanation:

Step1: Recall sampling - distribution properties

Statement I: The standard deviation of the sampling - distribution of the sample mean (also known as the standard error) is given by $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. As $n$ increases, $\sigma_{\bar{x}}$ decreases. So, the larger the sample, the smaller the standard deviation of the sampling distribution.

Step2: Central - location property

Statement II: The sampling distribution of the sample mean $\bar{X}$ has a mean $\mu_{\bar{X}}=\mu$, where $\mu$ is the population mean. So, sampling distributions of the mean will be centered at the mean of the population.

Step3: Central - Limit Theorem

Statement III: According to the Central - Limit Theorem, if we have a non - normal population with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample mean $\bar{X}$ (for samples of size $n$) is approximately normal when $n$ is large (usually $n\geq30$).

For Question #2, the mean of the sampling distribution of the sample mean $\mu_{\bar{X}}$ is equal to the population mean $\mu$. Given $\mu = 6.7$, the mean of the sampling distribution is 6.7.

Answer:

Question #1: A. I, II, and III
Question #2: D. 6.7