Two sample t-test

Independent sample t-test, two sample t-test
Difference of Two means Hypothesis test 등 같은 의미

가정

• 두 모집단 p1, p2 가 있다
• 각 집단에서 샘플을 취해서 그 평균을 모아 본다 (sampling distribution)

• p1의 샘플링분포 (distribution of sample means)

\begin{eqnarray*} & & \text {Assuming that X1 and X2 are independent } \\ \overline{X_{1}} & \sim & \left( \mu_{1}, \frac{\sigma_{1}}{n_{1}} \right) \\ \overline{X_{2}} & \sim & \left( \mu_{2}, \frac{\sigma_{2}}{n_{2}} \right) \\ & & \text{note that } n_{1}, n_{2} \text{ are sample size.} \\ & & \text{and } \frac{\sigma_{1}}{n_{1}} = Var \left[ \overline{X_{1}} \right] \\ Var \left[ \overline{X_{1}} - \overline{X_{2}} \right] & = & Var \left[ \overline{X_{1}} \right] + Var \left[ \overline{X_{2}} \right] \\ & = & \frac{\sigma_{1}}{n_{1}} + \frac{\sigma_{2}}{n_{2}} \\ \\ \text{SE}_{\overline{X_{1}} - \overline{X_{2}}} & = & \sqrt { \frac{\sigma_{1}}{n_{1}} + \frac{\sigma_{2}}{n_{2}} } \\ \text{SE}_{\text{diff}} & = & \sqrt { \frac{\sigma_{1}}{n_{1}} + \frac{\sigma_{2}}{n_{2}} } \\ \\ & & \text{If variances of each population } \\ & & \text{are the same, } \sigma_{1} = \sigma_{2} \\ & & \text{We use poooled variance, } \text{S}^{2}_{\text{p}}\\ \text{S}^{2}_{\text{p}} & = & \dfrac {\text{SS}_{1} + \text{SS}_{2}} {\text{df}_{1} + \text{df}_{2} } \\ & & \text{Hence, } \\ \text{SE}_{\text{diff}} & = & \sqrt {\frac{\text{S}^{2}_{\text{p}}}{n_1} + \frac{\text{S}^{2}_{\text{p}}}{n_2} } \\ \end{eqnarray*}