Formula

The general structure of confidence intervals for parameters

\[\Large \text{(Point Estimate)} \pm \text{(Margin of Error)}=\text{(Point Estimate)} \pm \text{(Critical Value)}\cdot \text{(Standard Error)}\]

The \(z\)-Confidence interval for a single population proportion (\(p\))

\[\color{red} {\Huge \hat{p}\pm z^*\cdot \sqrt{\frac{\hat{p}\cdot (1-\hat{p})}{n}}}\]

The \(t\)-Confidence interval for a single population mean (\(\mu\))

\[\color{red} {\Huge\bar{x}\pm t^*\cdot \frac{s}{\sqrt{n}}}\]

The \(z\)-Confidence interval for the difference between two population proportions (\(p_1 -p_2\))

\[\color{red} {\Huge (\hat{p}_1-\hat{p}_2)\pm z^*\cdot \sqrt{\frac{\hat{p}_1\cdot (1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2\cdot (1-\hat{p}_2)}{n_2}}}\]

The \(t\)-Confidence interval for the difference between two population means (\(\mu_1 -\mu_2\))

\[\color{red} {\Huge(\bar{x}_1-\bar{x}_2)\pm t^*\cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]

The The general structure of test statistics

\[\color{red} {\Huge\text{Test Statistic (t.s.)}=\frac{\text{Point Estimate} - \text{Null Value}}{\text{Standard Error}}}\]

The \(z\)-Test statistic for a single population proportion (\(p\))

\[\color{red} {\Huge z=\frac{\hat{p} - p_0}{\sqrt{p_0\cdot (1-p_0)/n}}}\]

The \(t\)-Test statistic for a single population mean (\(\mu\))

\[\color{red} {\Huge t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}}\]

The \(z\)-Test statistic for the difference between two population proportions (\(p_1 -p_2\))

\[\color{red} {\Huge z=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\frac{\hat{p}\cdot (1-\hat{p})}{n_1} + \frac{\hat{p}\cdot (1-\hat{p})}{n_2}}}}\] where \(\color{red} {\Large \hat{p}=\frac{\text{Total Number of Successes from Both Samples}}{\text{Total Sample Size}}}\), which is called the pooled sample proportion.

The \(t\)-Test statistic for the difference between two population means (\(\mu_1 -\mu_2\))

\[\color{red} {\Huge t=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}}\]

The \(\chi ^2\)-Test statistic for goodness of fit, independence, and homogeneity

\[\color{red} {\Huge \chi ^2 = \Sigma \frac{(\text{Observed}-\text{Expected})^2}{\text{Expected}}}\]