Question

consider a random variable 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.)

(b) if the mean of a random variable x is 30, what will be the mean of the sampling distribution of the sample mean?
$mu_{\bar{x}} = 30$

(c) as the sample size n increases, what happens to the standard error of the mean?
\\(\bigcirc\\) a. the standard error of the mean increases
\\(\bigcirc\\) b. the standard error of the mean remains the same.
\\(\checkmark\\) c. the standard error of the mean decreases.

(d) if the standard deviation of a random variable x is 20 and a random sample of size n = 21 is obtained, what is the standard deviation of the sampling distribution of the sample mean?
$sigma_{\bar{x}} = \square$ (type an exact answer, using radicals as needed )

Explanation:

Step1: Recall the formula for standard error (standard deviation of sampling distribution of sample mean)

The formula for the standard deviation of the sampling distribution of the sample mean (standard error) is $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.

Step2: Identify the given values

We are given that $\sigma = 20$ and $n = 21$.

Step3: Substitute the values into the formula

Substitute $\sigma = 20$ and $n = 21$ into the formula: $\sigma_{\bar{x}} = \frac{20}{\sqrt{21}}$. We can rationalize the denominator (though it's not necessary for an exact answer with radicals) or leave it as is.

Answer:

$\frac{20}{\sqrt{21}}$ (or equivalently $\frac{20\sqrt{21}}{21}$ after rationalizing, but the first form is also exact as per the question's request)