I want to test
$$H_0: \mu \leq 0 \,\,\,\,\,\, vs \,\,\,\,\,\, H_1: \mu > 0.$$ I am using a t test, so the statistic $T$ has $\nu$ degrees of freedom which depend on the sample size.
What is the difference between testing $H_0: \mu \leq 0$ or $H_0: \mu = 0$ vs $H_1: \mu >0$?
Is it possible to test only $H_0: \mu \leq 0$?
In R command
t.test(), it seems that this is equivalent to specifying
alternative = "greater", but there seems to be no distinction between the two hypothesis?