I'm trying to find the power function in the t test of two samples, (the variances are assumed to be equal ($\sigma_1=\sigma_2$ ), in the paper, on page 144 (5), I found that

"The power to detect a difference of $\delta=\mu_1-\mu_2$ with two-sided significance level $\alpha$ is given by:" $$1-\beta =T_v \left(t_{\alpha/2,v} ~ \left| \right. \frac{\delta}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}} \right)-T_v \left(-t_{\alpha/2,v} ~ |~\frac{\delta}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}} \right)$$

I tried to prove it, $1-\beta=P( |\bar{X_1}-\bar{X_2}|>cv ~|~\mu_1-\mu_2=\delta)$

$\beta=P(\frac{-cv-\delta}{s_p\sqrt{\sigma_1/n_1+\sigma_2/n_2}}<t<\frac{cv-\delta}{s_p\sqrt{\sigma_1/n_1+\sigma_2/n_2}})~~~~~cv =\mbox{critical value}$

what am I doing wrong? How can I prove it?

  • $\begingroup$ For the t test $\sigma$ = $\sigma_1$ = $\sigma_2$ is unknown and in the test statistic it is estimated by the pooled sample standard deviation.. $\endgroup$ – Michael Chernick Apr 28 '18 at 22:23

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