$H_0 : \beta_2 \le 0$ or $=0$

This is a part of my lecture note.

To test whether $\beta_2 \le 0$ at the 5% level of significance:

Hypothesis: $H_0 : \beta_2 =0$ and $H_1 : \beta_2 >0$

I don't understand $H_0: \beta_2 =0$ instead of $\beta_2 \le 0$.

My lecturer said it is because the worst case of $\beta_2$ is when it is equal to 0, but I don't really get what it means.

• I don't understand it as well. In a one-sided t-test $H_0$ could be that some parameter ist less or equal $0$. Maybe some context might help. Maybe we are in a situation where we only want to do two.sided testing? Maybe $\beta_2$ is a regression coefficient and our software performs two-sided tests on those and we want to use those? If so, please state in the question. – Bernhard Mar 14 '18 at 10:54
• The idea is that you will reject $H_0$ if the data show enough evidence for a high value of $\beta_2$. In this case you are interested to know if there is enough evidence that $\beta_2$ is larger than 0, which is the upper bound you want to check when testing if $\beta_2 \leq 0$. – Alessandro Mar 14 '18 at 11:45
• Bernhard yes it is a two sided test, and $\beta_2$ is a regression coefficient. Sorry for not mentioning it. – shk910 Mar 14 '18 at 22:41
• Alessandro Thanks, but why we can't say that $H_0: \beta_2 \le 0$ in this case?? My lecturer said $H_0$ must have equality only, but I don't understand. – shk910 Mar 14 '18 at 22:44
• Check this answer: stats.stackexchange.com/questions/7853/… and this: stats.stackexchange.com/questions/8196/… – Alessandro Mar 15 '18 at 11:13