Confidence Intervals

Definitions

Critical Value

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Critical Z Value

A value \(z_{\alpha}\) from the Standard Normal distribution is defined as,

\[P(\mathcal{Z} \leq z_{\alpha}) = 1 - \alpha\]

Critical T Value

A value \(t_{\alpha}\) from Student’s T Distribution is defined as,

\[P(\frac{\bar{\mathcal{X}}-\mu}{\mathcal{S}} \leq t_{\alpha}) = 1 - \alpha\]

Standard Error

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Standard Error

The standard error of an estimator \(\hat{\theta}\) is defined as,

\[s_{\theta} = E( (\theta - E(\hat{\theta}))^2 )\]

Margin of Error

Margin of Error

If c is a critical value from a point estimator \(\theta\)’s sampling distribution and \(s_{\theta}\) is the standard error of that estimator, then the margin of error for that point estimator is given by,

\[\text{MOE} = c \rvert \cdot s_{\theta}\]

Intervals

Confidence Intervals

If \(\theta\) is a population parameter, \(\hat{\theta}\) is a point estimator of \(\theta\) and \(\text{MOE}_{\theta}\) is the margin of error for that estimator, the confidence interval for \(\theta\) is given by,

\[\hat{\theta} - \text{MOE}_{\theta} \leq \theta \leq \hat{\theta} + \text{MOE}_{\theta}\]

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Population Proportion

Standard Error

Sample Proportion Standard Error

\[s_{\hat{p}} = \sqrt{ \frac{ \hat{p} \cdot (1-\hat{p}) }{n} }\]

Difference of Two Proportions

Difference of Sample Proportions Standard Error

\[s_{\hat{p}_1 - \hat{p}_2} = \sqrt{ \frac{ \hat{p}_1 \cdot (1-\hat{p}_1) }{n_1} + \frac{ \hat{p}_2 \cdot (1-\hat{p}_2) }{n_2} }\]

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Margin of Error

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Intervals

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Population Mean

Standard Error

Sample Mean Standard Error

\[\text{s}_{\bar{x}} = \frac{s}{\sqrt{n}}\]

Difference of Two Means

Difference of Means Standard Error

\[\text{s}_{\bar{x_2}_2 - \bar{x_1}} = \sqrt{ \frac{ s_{ \bar{x_1} } }{n_1} + \frac{ s_{ \bar{x_2}} }{n_2} }\]

Margin of Error

Population Mean Margin of Error, Known Standard Deviation

\[\text{MOE} = \lvert z_{1-\frac{\alpha}{2}} \rvert \cdot s_{\bar{x}}\]

Population Mean Margin of Error, Unknown Standard Deviation

\[\text{MOE} = \lvert t_{1-\frac{\alpha}{2}} \rvert \cdot s_{\bar{x}}\]

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Intervals

Confidence Interval For Population Mean

\[\bar{x} - \text{MOE} \leq \mu \leq \bar{x} + \text{MOE}\]

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Difference of Two Means

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Confidence Interval for Difference of Population Means

\[(\bar{x_2} - \bar{x_1}) - \text{MOE} \leq \mu_2 - \mu_1 \leq (\bar{x_2} - \bar{x_1}) + \text{MOE}\]

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