two_sample_t-test
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| two_sample_t-test [2026/04/07 06:42] – hkimscil | two_sample_t-test [2026/04/07 22:39] (current) – hkimscil | ||
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| [[:mean and variance of the sample mean]] 문서를 통해서 아래를 알고 있다. | [[:mean and variance of the sample mean]] 문서를 통해서 아래를 알고 있다. | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | \overline{X} & \sim & \left( \mu, \;\; \frac{\sigma}{n} \right) \\ | + | \overline{X} & \sim & \left( \mu, \;\; \frac{\sigma^2}{n} \right) \\ |
| & & \text{in other words, } \\ | & & \text{in other words, } \\ | ||
| E \left[ \overline{X} \right] & = & \mu \\ | E \left[ \overline{X} \right] & = & \mu \\ | ||
| - | Var \left[ \overline{X} \right] & = & \frac{\sigma}{n} \\ | + | Var \left[ \overline{X} \right] & = & \frac{\sigma^2}{n} \\ |
| & & \text {Assuming that X1 and X2 are independent } \\ | & & \text {Assuming that X1 and X2 are independent } \\ | ||
| - | \overline{X_{1}} & \sim & \left( \mu_{1}, \frac{\sigma_{1}}{n_{1}} \right) \\ | + | \overline{X_{1}} & \sim & \left( \mu_{1}, \frac{\sigma^2_{1}}{n_{1}} \right) \\ |
| - | \overline{X_{2}} & \sim & \left( \mu_{2}, \frac{\sigma_{2}}{n_{2}} \right) \\ | + | \overline{X_{2}} & \sim & \left( \mu_{2}, \frac{\sigma^2_{2}}{n_{2}} \right) \\ |
| & & \text{note that } n_{1}, n_{2} \text{ are sample size.} \\ | & & \text{note that } n_{1}, n_{2} \text{ are sample size.} \\ | ||
| & & \text{and } \\ | & & \text{and } \\ | ||
| - | & & \frac{\sigma_{1}}{n_{1}} = Var \left[ \overline{X_{1}} \right] \\ | + | & & \frac{\sigma^2_{1}}{n_{1}} = Var \left[ \overline{X_{1}} \right] \\ |
| \end{eqnarray*} | \end{eqnarray*} | ||
| 두 샘플 평균들의 차이를 모아 놓은 집합의 (distribution of sample mean difference) 성격은 아래와 같을 것이다. | 두 샘플 평균들의 차이를 모아 놓은 집합의 (distribution of sample mean difference) 성격은 아래와 같을 것이다. | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | E \left[ \overline{X_{1}} - \overline{X_{2}} \right] & = & | + | E \left[ \overline{X_{1}} - \overline{X_{2}} \right] & = & \mu_{1} - \mu_{2} \;, \;\;\; \text{and} \\ |
| - | \mu_{1} - \mu_{2} \;, \;\;\; \text{and} \\ | + | |
| Var \left[ \overline{X_{1}} - \overline{X_{2}} \right] & = & | Var \left[ \overline{X_{1}} - \overline{X_{2}} \right] & = & | ||
| Var \left[ \overline{X_{1}} \right] + Var \left[ \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}} | + | & = & \frac{\sigma^2_{1}}{n_{1}} + \frac{\sigma^2_{2}}{n_{2}} \\ |
| - | \\ | + | \text{SE}_{\overline{X_{1}} - \overline{X_{2}}} & = & \text{SE}_{\text{diff}} |
| - | \text{SE}_{\overline{X_{1}} - \overline{X_{2}}} & = & \text{SE}_{\text{diff}} = \sqrt { \frac{\sigma_{1}}{n_{1}} + \frac{\sigma_{2}}{n_{2}} } \\ | + | & = & \sqrt { \frac{\sigma^2_{1}}{n_{1}} + \frac{\sigma^2_{2}}{n_{2}} } \\ |
| \\ | \\ | ||
| & & \text{If variance of each population} \text{is unknown,} \\ | & & \text{If variance of each population} \text{is unknown,} \\ | ||
two_sample_t-test.1775544158.txt.gz · Last modified: by hkimscil