weird null hypothesis

Question

I've done loads of hypothesis tests for the mean where the null hypothesis is of the form $H_{0}=a$. Now I've come across one where the hypotheses are like

$H_{0}{\leq}a$ and $H_{1}>a$.

How does this affect the test statistic and critical value calculation?

Thanks

Answer 1

If you have a one-sided test (i.e. you have a direction, say greater than, for the inequality for the alternate hypothesis) then the null hypothesis $H_0: X = a$ is equivalent to the null hypothesis $H_0: X \leq a$. So the test remains the same.

Answer 2

Often it doesn't make a difference. However, in essence you are calculating the largest p-value under the null hypothesis. If the null is a single point, then its just the p-value of the null associated with that value. IF you have a range of values, then you want to find the highest p-value that can be produced by parameters in the set of null hypotheses. If this p-value is below your significance level, then you reject. Obviously, it is harder to reject the null when you allow it to have more freedom.