Does Google use the information supplied using Analytics to inform their search results?

If not, fair enough.

But if so, then might there be a case for not installing Google analytics?

If you install it, have a bounce rate of 90%, average visitor time of 3s, then Google are going to know this.

I think they might penalise a site which performs 'poorly' in GA, to the extent that it may well rank lower than a site which does not use GA.

tl;dr

Shitty sites with GA with be penalised to an extent even greater than sites without GA, due to the fact that sites with GA are telling Google exactly how shite they are, whereas it applies an 'average' shiteness to sites which it does not record metrics of.

They wont admit it but I would say they do yes!

I dont have GA on any un-subscribe pages or anything just incase really, ok not fool proof but it makes me feel better

But if you have a bounce rate of 90% and average of 3seconds per visitors then you need to sort that one out, GA is the least of your worrys!