Yes, you say "I don't think it is the case.", rather than "I think it is not the case.", if you think something is not the case.
That is, if 'think', 'suppose', 'believe' or 'imagine' is used to introduce a negative idea, you say 'I don't think...', 'I don't suppose...', etc. The usage is called 'transferred negation'.
You also say 'I don't think so.', 'I don't suppose so.', etc - in a negative 'short answer' structure.
The word 'hope', on the other hand, is an exception, for example, you say 'I hope they won't come.', not 'I don't hope they will come.'
So, you can try to make some sentences by using this structure - you can say "I don't think Expector knows all the grammar rules.", but not "I think Expector doesn't know all the grammar rules."
Hope it helps!