Section 7.1 Summary

1. The sampling distribution of the sample mean $\overline{x}$ for a given sample size $n$ consists of the collection of the means of all possible samples of size $n$ from the population. The mean of the sampling distribution of $\overline{x}$ is the value of the population mean $\mu$ (Fact 1). The standard error is ${\sigma }_{\overline{x}}=\sigma /\sqrt{n}$, where $\sigma$ is the population standard deviation (Fact 2). For a normal population, the sampling distribution of $\overline{x}$ is distributed as normal $\left(\mu ,\sigma /\sqrt{n}\right)$, where $\mu$ is the population mean and $\mu$ is the population standard deviation (Fact 3).
2. A simulation study showed that the sampling distribution of $\overline{x}$ for a skewed population achieved approximate normality when $n$ reached 30. The Central Limit Theorem is one of the most important results in statistics and is stated as follows: given a population with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample mean $\overline{x}$ becomes approximately normal $\left(\mu ,\sigma /\sqrt{n}\right)$ as the sample size gets larger, regardless of the shape of the population.
3. We can use Facts 3 and 4 to find probabilities for problems involving sample means.
4. Similarly, we can find percentiles for the sample means.