Test of difference between means

Let X1, · · · , Xn1 be a random sample from a N (μ1, σ2 1 ) distribution and Y1, · · · , Yn2 be a random sample from a N (μ2, σ2 2 ) distribution.
We want to test:

H_{0} : μ_{1} - μ_{2} = δ vs H_{a} : μ_{1} - μ_{2} \neq δ

MLEs of $μ_{1}$ and $μ_{2}$ are $\bar{x}$ and $\bar{y}$
MLEs of $δ = μ_{1} - μ_{2}$ is $\hat{δ} = \bar{x} - \bar{y}$

Known Variances

120

Unknown Variances

Case 1:
Sample sizes of $n_{1}, n_{2} \geq 30$

\begin{aligned} {\hat{σ}}_{1}^{2} = S_{1}^{2}, {\hat{σ}}_{2}^{2} = S_{2}^{2} \\ Z = \frac{\bar{x} - \bar{y} - δ_{0}}{\sqrt{\frac{S_{1}^{2}}{n_{1}} + \frac{S_{2}^{2}}{n_{2}}}} \sim N (0, 1) \\ \dots \end{aligned}

Case 2:
Either or both sample sizes are small ( $\leq 30$ )

Assume $σ_{1}^{2} = σ_{2}^{2} = σ^{2}$

\begin{aligned} {\hat{σ}}^{2} = S_{p o o l e d}^{2} = \frac{(n_{1} - 1) S_{1}^{2} + (n_{2} - 1) S_{2}^{2}}{n_{1} + n_{2} - 2} \\ t = \frac{\bar{x} - \bar{y} - δ_{0}}{\sqrt{\frac{{\hat{σ}}^{2}}{n_{1}} + \frac{{\hat{σ}}^{2}}{n_{2}}}} = \frac{\bar{x} - \bar{y} - {\hat{δ}}_{0}}{σ \sqrt{\frac{1}{n_{1}} + \frac{1}{n_{2}}}} \sim t_{n_{1} + n_{2} - 2} \end{aligned}