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T-test on paired means with unknown variance

Do two paired samples come from populations with the same mean $ \mu_0$ ?

Sometimes two samples are either generated or gathered as pairs of values $ \{(X_1, X_2)\}$ rather than as two separate samples $ \{X_1\}$ and $ \{X_2\}$, e.g. heights of twins. In this case, the two-sample test on means described above is inappropriate and a paired test has to be used. The paired test is based on testing the mean difference of all pairs $ D=X_1-X_2$ for zero mean.


$\displaystyle H_0:\mu_D$ $\displaystyle =$ 0 (6.7)
$\displaystyle H_1: \mu_D$ $\displaystyle \neq$ $\displaystyle 0 \nonumber$  

Test using a T-score test statistic with the sampling distribution
$\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{\overline{D}-\mu_0}{s_D/\sqrt{n}}\sim t_{n-1}$  


David Stephenson 2005-09-30