SESSION:7 The Central Limit TheoremFormula Review

Session:7 The Central Limit Theorem

Formula Review

Introductory Business Statistics | Leadership Development – Micro-Learning Session

Rice University 2020 | Michael Laverty, Colorado State University Global Chris Littel, North Carolina State University| https://openstax.org/details/books/introductory-business-statistics

7.1 The Central Limit Theorem for Sample Means

The Central Limit Theorem for Sample Means:

$\bar{X}$

$\bar{�}$ ~ N $(μ_{\bar{x}}, \frac{σ}{\sqrt{n}})$

$(�_{\bar{�}}, \frac{�}{\sqrt{�}})$

$Z = \frac{\bar{X} - μ_{\bar{X}}}{σ_{\bar{X}}} = \frac{\bar{X} - μ}{σ / \sqrt{n}}$

$� = \frac{\bar{�} - �_{\bar{�}}}{�_{\bar{�}}} = \frac{\bar{�} - �}{� / \sqrt{�}}$

The Mean $\bar{X} : μ_{\bar{x}}$

$\bar{�} : �_{\bar{�}}$

Central Limit Theorem for Sample Means z-score $z = \frac{\bar{x} - μ_{\bar{x}}}{(\frac{σ}{\sqrt{n}})}$

$� = \frac{\bar{�} - �_{\bar{�}}}{(\frac{�}{\sqrt{�}})}$

Standard Error of the Mean (Standard Deviation ( $\bar{X}$

$\bar{�}$ )): $\frac{σ}{\sqrt{n}}$

$\frac{�}{\sqrt{�}}$

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

$� = \frac{\bar{�} - �}{\frac{�}{\sqrt{�}} \cdot \sqrt{\frac{� - �}{� - 1}}}$

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