I've done loads of hypothesis tests for the mean (normal distribution) 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
The form you are describing is commonly referred to as a one-tailed test. As $\alpha$ only now refers to a single tail of the distribution, the critical value will change. For example, on a Z test, the critical value for $\alpha = .05$ on a two-tailed test is $\approx 1.96$, but for a one tailed-test with the same alpha the critical value would be $\approx 1.64$, i.e. the value where the one tail of the distribution contains $\alpha$ of the distribution.
Edit: Note that this answer only applies to tests where those $H_0$ and $H_1$s are sensible. There are contexts in which the critical value is drawn from a distribution where answering hypotheses in that form is not possible without additional work outside of the typical use of the statitic. For example, one would not easily use hypotheses of that form in relationship to an F test. See Why do we use a one-tailed test F-test in analysis of variance (ANOVA)? for more details on this disclaiming point.
