# Why are test hypotheses not symmetric?

Suppose, \begin{align} H_0\!: p<0 \\ H_1\!: p\geq0 \end{align} and further that I measure $-100$. Why am I not allowed to say that $H_0$ is true?

If you switch null and alternative hypotheses: \begin{align} H_0\!: p\geq0 \\ H_1\!: p<0 \end{align} and I measure $-100$, I can conclude that $p<0$.

I do not understand that by just reversing the hypotheses I can draw the conclusion according to what I measured.

The following case is clear to me: \begin{align} H_0\!: p=0 \\ H_1\!: p\neq0 \end{align} If you measure $0.1$ for example I understand you do not accept $H_0$, because $p$ could be $0.5$ or $-0.03$ etc. But if you measured $100$, you would reject $H_0$.

• You may want to use different notation or clarify what this parameter is supposed to be to avoid confusion. Is $p$ a probability? If so what does it mean to observe $-100$? In our last example how can $p = -0.03$? – dsaxton Oct 14 '16 at 15:01
• How would you calculate a probability under a null hypothesis such as that? You might want to look at stats.stackexchange.com/questions/8196/… – user20637 Oct 14 '16 at 15:11