Suppose I have $H_0: \beta = 0$ vs $H_1: \beta \neq 0$. My data tells me that I don't reject the null, with a p-value of approximately 0.06.
Then out of curiosity, since by the scientific theory one would expect $\beta<0$, I test $H_0: \beta \geq 0$ vs $H_1: \beta < 0$, and the data says that we should reject the null, with a p-value of 0.03...
I get the reason why we're having these two different conclusions has to do with the different quantiles we're using.
However, how should I conclude in this situation?