Question

The mean age of the employees at a large corporation is $35.2$ years, and the standard deviation is $9.5$ years.

A random sample of $4$ employees will be selected.

What are the mean and standard deviation of the sampling distribution of the sample mean for samples of size $4$ ?

• A. The mean is $35.2$, and the standard deviation is $9.5$.
• B. The mean is $35.2$, and the standard deviation is $\frac{9.5}{4}$.
• C. The mean is $35.2$, and the standard deviation is $\frac{9.5}{2}$.
• D. The mean is $\frac{35.2}{4}$, and the standard deviation is $\frac{9.5}{4}$.
• E. The mean is $\frac{35.2}{2}$, and the standard deviation is $\frac{9.5}{2}$.

Answer 1

The correct option is C

The mean is $35.2$, and the standard deviation is $\frac{9.5}{2}$.

Explanation for correct option:

For option $\left(C\right)$:

Finding the mean of the sampling distribution:

The mean of a sampling distribution according to the central limit theorem is equal to the mean of the population.
Therefore, mean of sampling distribution is $35.2$

Finding the Standard deviation of the sampling distribution:

Standard deviation $=\frac{\sigma }{\sqrt{n}}$
$$\begin{gathered} =\frac{9.5}{\sqrt{4}} \\ =\frac{9.5}{2} \end{gathered}$$
The mean is $35.2$, and the standard deviation is $\frac{9.5}{2}$.
Hence, option $\left(C\right)$ is correct.