Student's t-test

def. let $X_1,\dots ,X_n\stackrel{iid}{\sim }\mathcal{N}(\mu ,{\sigma }^2)$. We are comparing two hypothesis:

1. $H_0:\mu \le {\mu }_0$

2. $H_1:\mu >{\mu }_0$ Then, let $T=\frac{{\bar{X}}_n-{\mu }_0}{\hat{\sigma }/\sqrt{n}}$ where $\hat{\sigma }=\frac{E[({\bar{X}}_n-X_i)]}{n-1}$ is the besset-corrected variance estimator. We know that $T\sim t_{n-1}$. Thus the $\gamma =1-\alpha$-level student’s (=Gosset’s) t-test is:

$$\delta :\{\begin{array}{ll}{H}_0& \text{else }\\ H_1& \text{if }T>t_{n-1}(1-\alpha )\end{array}$$