## The Sampling Distribution of a Sample Mean

First, this section discusses the mean and variance of the sampling distribution of the mean. It also shows how central limit theorem can help to approximate the corresponding sampling distributions. Then, it talks about the properties of the sampling distribution for differences between means by giving the formulas of both mean and variance for the sampling distribution. Using the central limit theorem, it also talks about how to compute the probability of a difference between means being beyond a specified value.

### Sampling Distribution of the Mean

#### Questions

Question 1 out of 5.
The population has a mean of $14$ and a standard deviation of $3$. The sample size of your sampling distribution is $N=10$. What is the mean of the sampling distribution of the mean?

Question 2 out of 5.
The population has a mean of $30$ and a standard deviation of $6$. The sample size of your sampling distribution is $N=9$. What is the variance of the sampling distribution of the mean?

Question 3 out of 5.
The population has a mean of $120$ and a standard deviation of $12$. The sample size of your sampling distribution is $N=16$. What is the standard error of the mean?

Question 4 out of 5.
The sampling distribution of the mean, with $N=30$, of a moderately negatively skewed distribution is:

Positively skewed

Negatively skewed

Question 5 out of 5.
The entire student body of $225$ students took a test. These test scores have a mean of $75$, a standard deviation of $10$, and are slightly positively skewed. If you randomly chose $25$ of these test scores and calculated the mean over and over again, what could be the mean, standard deviation, and skew of this distribution?

Mean = $75$, SD = $10$, Skew = $1.2$

Mean = $75$, SD = $0.67$, Skew = $0.8$

Mean = $80$, SD = $2$, Skew = $-1.2$

Mean = $75$, SD = $2$, Skew = about $0$

Mean = $75$, SD = $0.67$, Skew = about $0$