## 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

1. The mean of the distribution of the difference between sample means is equal to the difference between population means. $20 - 15 = 5$
2. The variance of the sampling distribution of the difference between means is equal to the variance of the sampling distribution of the mean for Pop $1$ plus the variance of the sampling distribution of the mean for Pop $2$. $100/20 + 64/16 = 5 + 4 = 9$
3. Mean = $10$, SD = $4$, Plug these into the normal calculator and find the area above $6$. You get. $841$. A similar question using this data appears in the text.
4. Mean = $727 - 532$ = $195$, Var = $12,000/12 + 10,000/14$ = $1,714.3$, SD = $sqrt(1,714.3)$ = $41.404$, Use the normal calculator to calculate the area above $150$ for a distribution with this mean and SD. You get $0.8614$.