SECTION 5.2 SUMMARY

• • The population distribution of a variable is the distribution of its values for all members of the population.

• • The sample mean $\overline{x}$ of an SRS of size n drawn from a large population with mean μ and standard deviation σ has a sampling distribution with mean and standard deviation

  $$\begin{array}{lll}\hfill {\displaystyle {\mu }_{\overline{x}}}& {\displaystyle =}\hfill & {\displaystyle \mu }\hfill \\ {\displaystyle {\sigma }_{\overline{x}}}\hfill & {\displaystyle =}\hfill & {\displaystyle \frac{\sigma }{\sqrt{n}}}\hfill \end{array}$$

• The sample mean $\overline{x}$ is an unbiased estimator of the population mean μ and is less variable than a single observation. The standard deviation decreases in proportion to the square root of the sample size n. This means that to reduce the standard deviation by a factor of C, we need to increase the sample size by a factor of C².

• • The central limit theorem states that, for large n, the sampling distribution of $\overline{x}$ is approximately $N(\mu , \sigma /\sqrt{n})$ for any population with mean μ and finite standard deviation σ. This allows us to approximate probability calculations of $\overline{x}$ using the Normal distribution.

• • Linear combinations of independent Normal random variables have Normal distributions. In particular, if the population has a Normal distribution, so does $\overline{x}$.