When designing a test there is a list of Information about the test on the right, including availability dates, etc. But the panel does not indicate whether or not the test is published. To find this out, you need to go to the assignments tab, and then to change the status you need to click on the test and then click on edit. This is not only complicated and obscure. But it also frequently leads to the situation where instructors set up a test but don't realise it isn't published, and so the students don't see it.
Please add the published status (editable) to the Information on the Test design page.