Trouble understanding value given to null hypothesis in this problem

Question

My textbook has the following example problem:

The production manager of bullets claims less than 5% misfire. You have strong doubts about this claim. You decide to collect 1000 bullets and test how many misfire. Test whether the manager is correct at the $\alpha = 0.05$ level of significance.

The solution for this problem has the null hypothesis as $p = 0.05$ and the alternative hypothesis as $p > 0.05$.

My understand is that the null hypothesis represents a state of affairs that we will presume unless the sample data provides strong evidence to the contrary and the alternative hypothesis represents the claim for which we are seeking evidence from the data.

Given my understanding, It seems to me that the null hypothesis should be $p < 0.05$, since the claim is that less than $5\%$ misfire?

I would greatly appreciate it if people could please take the time to help me understand why the null hypothesis is not $p < 5\%$.

Answer 1

Most statistical tests are built to work with a point null hypotheses, i.e., $H_0\colon p=0.05$. However, if you test it against one-sided alternative $H_1\colon p>0.05$, usually you might as well include the remaining area into the null and test $H_0\colon p\leq 0.05$. For such a pair $H_0$-$H_1$ $p=0.05$ is the least favourable configuration of the null, because it is the closest one to the alternative. That's why you could calculate p-value for $H_0\colon p=0.05$ and use it to test $H_0\colon p\leq 0.05$.