================================================================================ - Not enough sample - Sample populations variance ================================================================================ Required assumption - 2 populations should have "normal distribution" or "similar normal distribution" $$$\mathcal{N}(\mu_A,\sigma_A^2)$$$ $$$\mathcal{N}(\mu_B,\sigma_B^2)$$$ $$$\sigma_A^2=\sigma_B^2=\sigma^2$$$ $$$s_A^2=s_B^2=s^2$$$ ================================================================================ Pooled variance estimator $$$S_p^2$$$ - There are populations - They are independent - Pooled variance estimator is expressed by "unbiased estimator of comman variance" ================================================================================ $$$n$$$: number of populations $$$n_i$$$: sample size ================================================================================ In both sided test, confidence interval $$$(\bar{x}_A - \bar{x}_B) - t_{(n_A+n_B-2,\frac{\alpha}{2})} S_p \sqrt{\frac{1}{n_A}+\frac{1}{n_B}} \le \mu_A-\mu_B \le (\bar{x}_A - \bar{x}_B) + t_{(n_A+n_B-2,\frac{\alpha}{2})} S_p \sqrt{\frac{1}{n_A}+\frac{1}{n_B}}$$$ ================================================================================ Let's establish hypothesis test statistic t ================================================================================ Case - Not enough sample - No information about variance - That is, you don't know variance uniformity of populations $$$\bar{x}_{A} - \bar{x}_{B}$$$ $$$E(\bar{x}_A - \bar{x}_B) = E(\bar{x}_A) - E(\bar{x}_B) = \mu_A-\mu_B$$$ $$$s^2(\bar{x}_A - \bar{x}_B) = s^2(\bar{x}_A) + s^2(\bar{x}_B) - 2Cov(\bar{x}_A-\bar{x}_B) = \frac{s_{x_A}^2}{n_A} + \frac{s_{x_B}^2}{n_B}$$$ ================================================================================ Confidence interval (when variance uniformity is unknown) ================================================================================ Establish hypothesis and create statistic t