Is there a significance test for testing that the population standard deviation is greater than zero?

Question

I am working on a project where I need to test the following hypotheses:
$H_0: σ = 0 H_a: σ > 0$
I am aware of the chi square test for standard deviation, that is
$χ^2 = \frac{(n-1)s^2}{σ^2}$
However, as σ = 0, this test statistic would result in me dividing by 0. Is there any way for me to test for the standard deviation being greater than 0?

Answer 1

I'm fairly sure it shapes up like this: if the population standard deviation is zero, then any sample you take from it has to be all the same value, because that's the only value with nonzero probability. So if you have a sample with more than $1$ distinct value, you can conclude with $p = 0$ that the population standard deviation is nonzero, because the sample you got is completely impossible if the null hypothesis is true. If you have a sample with all the same value, then because the sample standard deviation can't be negative and must always be at least $0,$ with $p = 1$ we fail to reject the null hypothesis, because we failed to collect sufficient evidence that the population standard deviation is nonzero.