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.) ​
(a) If a random variable X is normally​ distributed, what will be the shape of the distribution of the sample​ mean?
a. Normal
b. Skewed right
c. Skewed left
d. Cannot be determined
​(b) If the mean of a random variable X is 40​, what will be the mean of the sampling distribution of the sample​ mean? mu Subscript x overbarequals 40
​(c) As the sample size n​ increases, what happens to the standard error of the​ mean?
A. The standard error of the mean increases
B. The standard error of the mean remains the same.
C. The standard error of the mean decreases.
(d) If the standard deviation of a random variable X is 10 and a random sample of size nequals15 is​ obtained, what is the standard deviation of the sampling distribution of the sample​ mean? sigma Subscript x overbarequals nothing ​(Type an exact​ answer, using radicals as​ needed.)

Answer 1

Answer:

a)  a. Normal

b) The mean of the sampling distribution of the sample mean is 40.

c) C. The standard error of the mean decreases.

d) The standard deviation of the sampling distribution of the sample​ mean is 2.58.

Step-by-step explanation:

(a) If a random variable X is normally​ distributed, what will be the shape of the distribution of the sample​ mean?

If the variable is normally distributed, then the sample mean is going to be normally distributed. So the correct answer is

a. Normal

​(b) If the mean of a random variable X is 40​, what will be the mean of the sampling distribution of the sample​ mean?

The mean of the sampling distribuion is the same, so it is 40.

​(c) As the sample size n​ increases, what happens to the standard error of the​ mean?

We have that:

$S_{E} = \frac{\sigma}{\sqrt{n}}$

In which $\sigma$ is the standard deviation. So, as the sample size n increases, the standard error of the mean decreases. The correct answer is

C. The standard error of the mean decreases.

(d) If the standard deviation of a random variable X is 10 and a random sample of size nequals15 is​ obtained, what is the standard deviation of the sampling distribution of the sample​ mean

We have that:

$s = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{15}} = 2.58$

The standard deviation of the sampling distribution of the sample​ mean is 2.58.