Huge, gaping disconnect in the blame for the death (spoilers)

They needed the girl to be committed to the asylum for killing her little sister for this to work. So they go through this overt effort to have the "monster" convince her that she killed her own little sister. That she was jealous of her and wanted to go back to being the only child.

The problem with this is that she never told the cops any of it. She insisted the entire time that the monster killed her. How did they arrive at the conclusion that she killed her own sister?

Here are the facts as presented to the viewer
> Younger sister falls in pool and dies
> Older sister insists the monster killed her
> It's Halloween. There are plenty of people dressed up as monsters

How do the cops then conclude that she killed her own sister? This is so stupid.

I get it. They put that dialog in there for us, the viewers, to make this connection - that she is jealous, killed her, and was committed to the asylum for this reason. But within the story it all falls apart. Actually, it doesn't fall apart because it doesn't ever even connect to begin with.

Why would the cops make this assumption at all? It could have been an accident. It could have been a "monster" on Halloween according to the girl's testimony. But to make the leap to sororcide for no reason, without any evidence or indication? Soooooo stupid. Ack.