Is there something odd about the way statistics are used?

From the little that I know about statistics there seems to be something not quite right in the way they are used. For example, suppose that a poll is taken to find out whether people are in favor of a certain issue. A sample is taken and it is found that 30% of the sample backed the issue.

The statistician will now give a confidence interval for the results. Let’s say that in the present case that the 95% confidence interval is plus or minus 3%. You have to understand that this does not mean that there is a 95% chance that the actual percentage is between 27% and 33%. What it means is that if the actual percentage is between 27% and 33% then there would be at least a 95% chance of getting our particular results. This seems to be doing things backward.

By way of analogy, suppose a farmer is trying to figure out why his pig got out of its pen. He might reason that if the pig was able to fly there would be a 100% chance of it escaping, so he adopts that as the explanation.

I am not suggesting that we stop using statistics, just saying that there is something about the whole idea behind the way it is used that I find a little unsettling.