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Mean Value: \(E(\hat{\theta})\)
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\[p\]
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\[\mu\]
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\[\mu\]
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\[p_1 - p_2\]
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\[\mu_1 - \mu_2\]
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\[\mu_d\]
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Variance: \(\text{Var}(\hat{\theta})\)
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\[\frac{p(1-p)}{n}\]
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\[\frac{\sigma^2}{n}\]
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\[\frac{\sigma^2}{n}\]
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\[\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}\]
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\[\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}\]
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\[\frac{\sigma_d^2}{n}\]
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Standard Error: \(SE\)
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\[\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]
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\[\frac{\sigma}{\sqrt{n}}\]
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\[\frac{s}{\sqrt{n}}\]
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\[\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\]
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\[\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}\]
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\[\frac{S_d}{\sqrt{n}}\]
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Pivotal Quantity (CI)
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\[\frac{\hat{p} - p}{\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}}\]
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\[\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}\]
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\[\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}\]
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\[\frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}\]
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\[\frac{(\bar{y}_1 - \bar{y}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}}\]
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\[\frac{\bar{d} - \mu_d}{\frac{S_d}{\sqrt{n}}}\]
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Test Statistic
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\[\frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]
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\[\frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}\]
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\[\frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}\]
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\[\frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}\]
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\[\frac{(\bar{y}_1 - \bar{y}_2) - \Delta_0}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}}\]
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\[\frac{\bar{d} - \mu_{d0}}{\frac{S_d}{\sqrt{n}}}\]
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Distribution Model
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\[N(0,1)\]
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\[N(0,1)\]
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\[t_{n-1}\]
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\[N(0,1)\]
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\[t_{\nu}\text{ with } \nu = \frac{\left(\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}\right)^2}{\frac{\left(\frac{S_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{S_2^2}{n_2}\right)^2}{n_2-1}}\]
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\[t_{n-1}\]
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Margin of Error: \(ME\)
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\[z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]
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\[z^* \frac{\sigma}{\sqrt{n}}\]
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\[t^* \frac{s}{\sqrt{n}}\]
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\[z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\]
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\[t^* \sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}\]
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\[t^* \frac{S_d}{\sqrt{n}}\]
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