I think the answers and comments posted here are somewhat not complete about how -시죠/시지요 is used. It does mark questions in some circumstances, but it can also act as a highly formal suggestion. 주무시지요 means almost always "Please go to bed", not "Are you sleeping?" since a sleeping person is not expected to answer that. On contrary, 지루하시죠 always means "You are bored, aren't you?" not a suggestion to get bored because such suggestion is absurd and not anything achievable voluntarily.
While -시지요 itself can be either question or suggestion, and in most cases you need research as @Nontofull pointed out, it is clearly a question in this context, because of the negation, 안.
Though I'm not a linguist, let me demonstrate in my ability.
- 강아지 키우시지요
- 강아지 안 키우시지요
- 강아지 키우지 마시지요
(BTW, 강아지 is more apt to be translated into puppy rather than dog, which is 개.)
1 can be interpreted in both ways; it can be interpreted as "Do you raise a puppy?" or "I suggest you to raise a puppy." and it requires intonation and context to be understood correctly.
2 can only be understood as a question, since 안 means somewhat descriptive on the situation. You cannot suggest someone to "currently not raise a puppy." It is a strict question.
3 can only be understood as a suggestion, since 마-, or dictionary form 말다, means forbid. You cannot ask someone if they forbade to raise a puppy (it makes some sense in English, but it's because of translation and this is not even remotely making any sense in Korean for this to be a question).
So there you have it. "강아지 안 키우시죠" is translated as "You don't have a dog, do you?" If you see 안 -시죠 at the end of the sentence, I can say it is almost certain (I can't think of a counterexample.) it is a question, not a suggestion.