I had to link to an excellent post at Sisu Hockey looking at shot quality and number of shots.
A couple interesting things: For shots on offence, it is not just the number of shots that matters. The type of shots taken matters a great deal, as goals for correlated just as well with expected goals (which is based on a shot quality calculation) as with simple shot totals.
On defence, however, the correlation between shots allowed and goals was just as high as it was between expected goals against and goals. This means that there is no general rule that teams that allow more shots also allow lower shot quality, or that teams that allow fewer shots allow more difficult chances. In general, bad teams allow more shots and more dangerous scoring chances. So goalies on good teams have a big advantage.
Another interesting stat is that in the first round of this year's playoffs, the team that was expected to score more goals won. There were a few games where a strong or weak performance by a goalie made the difference, but over the course of 7 games or less, it was the shooters' results that carried the day every time. If you switched goalies in all the first round series, probably the same teams would win. Just another thing to think about when someone on TV talks about how this or that series will come down to the goaltending matchup.