Part 4 of 4
0 of 1 Point
Consider a random variable $\mathrm{X}$ that is normally distributed. Complete parts (a) through (d) below.
(This is a reading assessment question. Be certain of your answer because you only get one attempt on this question.)
\[
\mu_{x}^{-}=30
\]
(c) As the sample size n increases, what happens to the standard error of the mean?
A. The standard error of the mean decreases.
B. The standard error of the mean remains the same.
C. The standard error of the mean increases
(d) If the standard deviation of a random variable $X$ is 10 and a random sample of size $n=21$ is obtained, what is the standard deviation of the sampling distribution of the sample mean?
$\sigma_{x}=\square$ (Type an exact answer, using radicals as needed)

Step 1 ：For part (c), the standard error of the mean is defined as the standard deviation divided by the square root of the sample size. As the sample size increases, the denominator of this fraction becomes larger, which makes the overall fraction smaller. Therefore, the standard error of the mean decreases as the sample size increases. So, the correct answer is A. The standard error of the mean decreases as the sample size increases.

Step 2 ：For part (d), the standard deviation of the sampling distribution of the sample mean, also known as the standard error, is calculated as the standard deviation divided by the square root of the sample size. Given a standard deviation of 10 and a sample size of 21, we can substitute these values into the formula to find the standard error.

Step 3 ：Using the formula for standard error, \(\sigma_{x} = \frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the standard deviation and \(n\) is the sample size, we substitute the given values to get \(\sigma_{x} = \frac{10}{\sqrt{21}}\).

Step 4 ：Solving the above expression, we get \(\sigma_{x} = 2.182178902359924\).

Step 5 ：So, the standard deviation of the sampling distribution of the sample mean is \(\boxed{2.182178902359924}\).