# 5.6: Chapter 7 Formula Review

## 7.1 The Central Limit Theorem for Sample Means

The Central Limit Theorem for Sample Means:

$$\overline{X} \sim N\left(\mu_{\overline{x}}, \frac{\sigma}{\sqrt{n}}\right)$$

$$Z=\frac{\overline{X}-\mu_{\overline{X}}}{\sigma_{X}}=\frac{\overline{X}-\mu}{\sigma / \sqrt{n}}$$

The Mean $$\overline{X} : \mu_{\overline x}$$

Central Limit Theorem for Sample Means z-score $$z=\frac{\overline{x}-\mu_{\overline{x}}}{\left(\frac{\sigma}{\sqrt{n}}\right)}$$

Standard Error of the Mean (Standard Deviation $$(\overline{X}) ) : \frac{\sigma}{\sqrt{n}}$$

Finite Population Correction Factor for the sampling distribution of means: $$Z=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}} \cdot \sqrt{\frac{N-n}{N-1}}}$$

Finite Population Correction Factor for the sampling distribution of proportions: $$\sigma_{\mathrm{p}^{\prime}}=\sqrt{\frac{p(1-p)}{n}} \times \sqrt{\frac{N-n}{N-1}}$$