## 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 Difference Between Means

#### Questions

Question 1 out of 4.
Population $1$ has a mean of $20$ and a variance of $100$. Population $2$ has a mean of $15$ and a variance of $64$. You sample $20$ scores from Pop $1$ and $16$ scores from Pop $2$. What is the mean of the sampling distribution of the difference between means (Pop $1$ - Pop $2$)?

Question 2 out of 4.
Population $1$ has a mean of $20$ and a variance of $100$. Population $2$ has a mean of $15$ and a variance of $64$. You sample $20$ scores from Pop $1$ and $16$ scores from Pop $2$. What is the variance of the sampling distribution of the difference between means (Pop $1$ - Pop $2$)?

Question 3 out of 4.
The mean height of $15$-year-old boys is $175$ cm and the variance is $64$. For girls, the mean is $165$ and the variance is $64$. If $8$ boys and $8$ girls were sampled, what is the probability that the mean height of the sample of boys would be at least $6$ cm higher than the mean height of the sample of girls?

Question 4 out of 4.
The mean time to complete a task is $727$ millisecond for 3rd graders and $532$ milliseconds for 5th graders. The variances of the two grades are $12,000$ for 3rd graders and $10,000$ for 5th graders. The times for both grades are normally distributed. You randomly sample $12$ 3rd graders and $14$ 5th graders. What is the probability that the mean time of the 3rd graders will exceed the mean time of the 5th graders by $150$ msec or more?