Power (statistics)
Analytic solution

Thanks to t-test theory, we know this test statistic under the null hypothesis follows a Student t-distribution with n − 1 {\displaystyle n-1} degrees of freedom. If we wish to reject the null at significance level α = 0.05 {\displaystyle \alpha =0.05\,}, we must find the critical value t α {\displaystyle t_{\alpha }} such that the probability of T n > t α {\displaystyle T_{n}>t_{\alpha }} under the null is equal to α {\displaystyle \alpha }. If n is large, the t-distribution converges to the standard normal distribution (thus no longer involving n) and so through use of the corresponding quantile function Φ − 1 {\displaystyle \Phi ^{-1}}, we obtain that the null should be rejected if
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