7.8: Formula Review

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7.2: The Central Limit Theorem for Sample Means

The Central Limit Theorem for Sample Means:

\[\begin{array}{l}
\bar{X} \sim N\left(\mu_{\bar{x}}, \dfrac{\sigma}{\sqrt{n}}\right) \\
Z=\dfrac{\bar{X}-\mu_{\bar{X}}}{\sigma_{\bar{X}}}=\dfrac{\bar{X}-\mu}{\sigma / \sqrt{n}}
\end{array}\]

The Mean \(\bar{X}: \mu_{\bar{x}}\)

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

Standard Error of the Mean (Standard Deviation \((\bar{X})\) ): \(\dfrac{\sigma}{\sqrt{n}}\)

Finite Population Correction Factor for the sampling distribution of means: \(Z=\dfrac{\bar{x}-\mu}{\dfrac{\sigma}{\sqrt{n}} \cdot \sqrt{\dfrac{N-n}{N-1}}}\)

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

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7.2: The Central Limit Theorem for Sample Means