Interval estimation for the difference between means

Examples

$X_{1}, . ., X_{n_{1}} \sim N (μ_{X}, σ_{X}^{2}), Y_{1}, \dots, Y_{n_{2}} \sim N (μ_{y}, σ_{Y}^{2})$
If $σ_{X}^{2}, σ_{y}^{2}$ are known:

\begin{aligned} {\hat{μ}}_{X} = \bar{x}, {\hat{μ}}_{Y} = \bar{y} \\ \bar{x} \sim N (μ_{x}, \frac{σ_{x}^{2}}{n_{1}}), \bar{y} \sim N (μ_{Y}, \frac{σ_{Y}^{2}}{n_{2}}) \\ \bar{x} - \bar{y} \sim N (μ_{X}, μ_{Y}, \frac{σ_{X}^{2}}{n_{1}} + \frac{σ_{Y}^{2}}{n_{2}}), d = μ_{X} - μ_{Y} \\ Z = \frac{\bar{x} - \bar{y} - D}{\sqrt{\frac{σ_{X}^{2}}{n_{1}} + \frac{σ_{Y}^{2}}{n_{2}}}} \sim N (0, 1) \\ P (D_{l} \leq D \leq D_{u}) = 1 - α \\ P (\frac{\bar{x} - \bar{y} - D_{u}}{\sqrt{\frac{σ_{X}^{2}}{n_{1}} + \frac{σ_{Y}^{2}}{n_{2}}}} \leq \frac{\bar{x} - \bar{y} - D}{\sqrt{\frac{σ_{X}^{2}}{n_{1}} + \frac{σ_{Y}^{2}}{n_{2}}}} \leq \frac{\bar{x} - \bar{y} - D_{l}}{\sqrt{\frac{σ_{X}^{2}}{n_{1}} + \frac{σ_{Y}^{2}}{n_{2}}}}) = 1 - α \\ C I for D = μ_{X} - μ_{Y} : () \end{aligned}

$σ_{X}^{2}, σ_{Y}^{2}$ are unknown:

\begin{aligned} {\hat{σ}}_{X}^{2} = S_{X}^{2} = \sum_{i = 1}^{n_{1}} \frac{(X_{i} - \bar{X})^{2}}{n - 1} \\ {\hat{σ}}_{y}^{2} = S_{y}^{2} = \sum_{i = 1}^{n_{2}} \frac{(Y_{i} - \hat{Y})^{2}}{n_{2} - 1} \\ for n_{1} \geq 30, n_{2} \geq 30 replace σ^{2} s by S^{2} s \\ CI for D = μ_{x} - μ_{y} : \\ (\bar{x} - \bar{y} - Z_{\frac{α}{2}} \sqrt{\frac{S_{x}^{2}}{n_{1}} + \frac{S_{Y}^{2}}{n_{2}}}, \bar{x} - \bar{y} + Z_{\frac{α}{2}} \sqrt{\frac{S_{x}^{2}}{n_{1}} + \frac{S_{Y}^{2}}{n_{2}}}) \\ for small n_{1}, n_{2} < 30, we pool information from the 2 samples \\ {\hat{σ}}_{p o o l e d}^{2} = S_{p o o l e d}^{2} = \frac{\sum^{n_{1}} (x_{i} - \bar{x})^{2} + \sum^{n_{2}} (y_{i} - \bar{y})^{2}}{n_{1} + n_{2} - 2} \\ = \frac{(n_{1} - 1) S_{x}^{2} + (n_{2} - 1) S_{y}^{2}}{n_{1} + n_{2} - 2} \\ t & = \frac{\bar{x} - \bar{y} - D}{\sqrt{V \hat{a} r (\bar{x} - \bar{y})}} = \frac{\bar{x} - \bar{y} - D}{\sqrt{\frac{{\hat{σ}}_{X}^{2}}{n_{1}} + \frac{{\hat{σ}}_{Y}^{2}}{n_{2}}}} = \frac{\bar{x} - \bar{y} - D}{\sqrt{\frac{S_{p o o l e d}^{2}}{n_{1}} + \frac{S_{p o o l e d}^{2}}{n_{2}}}} \\ = \frac{\bar{x} - \bar{y} - D}{S_{p o o l e d} \sqrt{\frac{1}{n_{1}} + \frac{1}{n_{2}}}} \sim t_{n_{1} + n_{2} - 2} \\ CI for D = μ_{x} - μ_{y} : \\ (\bar{x} - \bar{y} - t_{1 - \frac{α}{2}, n_{1} + n_{2} - 2} \sqrt{S_{p o o l e d}^{2} (\frac{1}{n_{1}} + \frac{1}{n_{2}})}, \bar{x} - \bar{y} + t_{1 - \frac{α}{2}, n_{1} + n_{2} - 2} \sqrt{S_{p o o l e d}^{2} (\frac{1}{n_{1}} + \frac{1}{n_{2}})}) \end{aligned}

Test of Means