Occasional reflections on Life, the World, and Mathematics

According to The Guardian, people want Boris Johnson to be the next leader of the Conservatives. They don’t say it explicitly, but they suggest that “next” means, like, tomorrow, and not after the next election. After citing a poll finding that 29% of voters want Johnson to be the next Tory leader (are those Conservative supporters? I might want Johnson to be the next Tory leader because I think he’ll lead his party to disaster…), they write

The strong support for Johnson feeds into the party standings. The poll finds that Labour’s seven-point lead would fall to three points if he led the Tories. The Tories would see their support increase by three points under a Johnson premiership to 34% while Labour would see its support fall by one point to 37%. Johnson would also hit support for Ukip,. which would see its support fall by two points to 8%.

Before the Tories dump Cameron, they might want to check whether this 3% boost is statistically robust. This looks like an elementary statistics exercise, but it’s not quite so simple. If D is the Tory support under Cameron, and B the Tory support under Johnson, then B-D might be expected to be about 3%. But how confident should we be that Johnson is really better than Cameron? Unfortunately, we can’t know that without knowing the correlations: in this case, that means we need to know how many people supported the Tories only with Cameron, and how many supported them only with Johnson, and how many supported them with either leader.

At one extreme, with maximum overlap, 31% support the Cameron Tories and the same people support the Johnson Tories, and there are 3% extra JT supporters. In that case, since we sampled 1002 voters and found no CT voters, we can be pretty certain that there aren’t any of them. (We can be 97.5% sure that it’s under 0.4%.) The fraction of JT voters is >2% with 97.5 % confidence. So we have a better than 95% confidence interval that there are at least 1.6% more exclusive JT supporters than exclusive CT supporters. (And less than 2 in a billion chance that there is actually more CT support than JT support.)

At the other extreme, the JT voters are a completely different set of voters than the CT voters. Then there are 31% CT supporters, 34% JT supporters, and 35% who wouldn’t vote for the Tories even if William Pitt the Younger rose from the grave to lead them. Then we have a perfectly standard first-year statistics question: The difference between JT and CT then has expectation 3%, with standard error 2.5% (= sqrt(.65/1002)). So a 95% confidence interval for the difference is about (-2%,8%), and I wouldn’t bet much on improving Tory chances by switching leaders. (Specifically, there would be about a 12% chance of seeing this result if the two leaders have equal support.)

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