How to calculate test statistic when $H_0$ is not equal

Question

So I'm trying to perform a $t$ test with sample mean $\bar{x}$ and sample variance $s^2$. I know how to do this with $H_0 = a$ and $H_a \ne a$. But now I'm given $H_0 < a$ and $H_a \ge a$. How do I modify the $t$ test formula?

Answer 1

Normally, you should use one tailed test. When $H_0 : \mu<a$, it is not strange that $t=\frac{\overline{x}-a}{\sqrt{\frac{s^2}{n}}}$ is negative. So, even when $t$ is $-500$, $H_0$ is not rejeted. However, t is larger than $t_\alpha$(not $t_{\frac{\alpha}{2}}$), it is rejected.