At a school the pass rate in 'A' level mathematics was $40\%$. After a new teacher was hired, $4$ out of $9$ students passed. A test is to be run at the $5\%$ level for evidence of an improvement. Which pair of hypotheses is correct?

The null hypothesis always states the value being assumed until the data argue otherwise, which here is the historical pass rate $p = 0.4$, and it is always an equality. The question asks specifically about an improvement, so the alternative is one-sided, $p > 0.4$. The observed proportion $\dfrac49$ is the evidence, not the hypothesis.