Sampling and Confidence Intervals

Sampling distributions

Sample mean

μ_{\bar{x}} = μ σ_{\bar{x}} = \frac{σ}{\sqrt{n}} S E_{\bar{x}} = \frac{s}{\sqrt{n}}

Sample proportion

μ_{\hat{p}} = p σ_{\hat{p}} = \sqrt{\frac{p (1 - p)}{n}} S E_{\hat{p}} = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}

z-interval for a proportion

\hat{p} \pm z^{*} \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}

t-interval for a mean

\bar{x} \pm t^{*} \frac{s}{\sqrt{n}}

df = n − 1

Chi-square statistic

χ^{2} = \sum \frac{(observed - expected)^{2}}{expected}

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