This is a very interesting question, because proving that a ghost "really exists" is very controversial in our modern society, leading to such reality TV as Paranormal State and fictional TV such as Ghost Whisperer. What drives these shows is the beliefs of those who think they can see and track ghosts, and sometimes the conflict appears between those who can see struggling with those who can't. So, let's analyze Act 3, scene 4, from that modern perspective: who can see the ghost of Banquo, and who can't? It's not that we have to say that ghosts are truly real as much as we have to prove that ghosts are making a difference to the characters and plot in this story.
There's an obvious first answer, and you already know it: Macbeth believes the ghost really exists. Why? Because Macbeth takes concrete action in response to the ghost. Go back to the scene and find the lines of dialogue where Macbeth a) speaks to the ghost b) speaks about the ghost and c) takes physical action in response to the ghost. (Hint for your first piece of evidence, relating to (c): what is Macbeth unable to do as the banquet starts?)
Now, besides Macbeth, who else gets to see the ghost? Remember, this story is drama, so it's the audience. Where can you find evidence that the audience gets to see the ghost? When you analyze literature, you can quote anything on the page that proves your point. Ever been in a play? Then you know that stage directions are key (written in italics) and without them, actors, such as the one playing the ghost of Banquo, wouldn't know what Shakespeare wanted them to do).
Keep in mind: you as writer are acting like the college students on Paranormal State: you're using any kind of "tracking devices" (EVP tracking equipment, audio equipment,and video equipment with infrared/temperature tracking).
Finally, if you really want to dunk on this one, show your teacher that the play up until this point has given the audience ample proof that the spiritual world is alive and well in this play. Where in Acts 1-3, up to this moment, have we seen evidence of a) supernatural phenomena and b) characters reacting to it? If there is evidence for that, then why should a ghost at this point be unbelievable? And there you have your evidence.
Good luck, counselor! Now go win your case in court. :-)
Does the question you've been set give you any opportunity to put the opposite point of view, that the ghost 'really exists' only in Macbeth's mind? The previous answer gave you good ammunition in terms of characters, actors and audience, but many modern productions of the play prefer to play the scene without Banquo's ghost actually appearing. Film, of course, is ideal for this - in Roman Polanski's terrifying version the 'blood-boltered Banquo' is a grinning hallucination.
What's terrifying - and tragic - is the disintegration of Macbeth's personality under the weight of guilt. We know from later dialogue that he continues to have nightmares about both Duncan and Banquo, and Lady Macbeth's sleeping walking scene shows that they are both in mental turmoil over all these murders, and that of Lady Macduff - 'The thane of Fife had a wife - where is she now?'
Shakespeare probably believed in ghosts - as did his audience - but we will never know. The evidence of his plays, however, is that he believed 100% in the corrosive power of guilt, and in the natural justice that brings Macbeth to his downfall and death.
Challenge the question - what have you got to lose? Good luck!