Hold on for the Ministry job. Avoid answering the Stat board offer as long as possible.

You can try speaking with the HR at the Ministry to get further comfort.

If it comes to the point where you have to give up the Stat board offer without getting a formal offer from the Ministry, give it up.

Personally, I will regret missing out on my first choice and ending up with the 2nd more than if I ended up with none.

Just like relationships, you wait for the one you want and not pick the 2nd best just because he offers you first right? (not untill you are 30