What role do confidence intervals play in politics?
One role would be to determine whether the figures from a constituent poll are worth acting on.
In politics, all statistics are generally just abused. Confidence intervals of anything simply tell you how wide a range of values could be just happenstance – i.e. NOT statistically ‘significant’.
In polls, when you see things like a “55-45 split/lead, with a 10% confidence interval” it really means that the data sample shows a slight difference, but due to the sample size, the “confidence” in the observed difference could completely invert the ratio.
Most people – maybe particularly politicians – really do NOT understand statistics or how to use them correctly. As long a they can sound “scientific” and make their side look good, they’ll go with it.