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Simultaneous confidence intervals
How do we show plausible ranges for many estimates at once?
Simultaneous confidence intervals have many applications, including in quality control and opinion polling. The Royal Statistical Society commissioned a survey of MPs, with Savanta ComRes asking:
If you toss a fair coin twice, what is the probability of getting two heads?
As in the previous survey in 2011 with Ipsos MORI, there were seven response options. Parliamentarians could choose: 15%, 25%, 40%, 50%, 75%, ‘Other’ (with a box to input) or that they did not know.
In many analyses, there are binary outcomes: yes or no, in or out, Manchester City or Manchester United, and so on. For that case, we have a Binomial distribution. That distribution models the number of ‘successes’ or ‘failures’.
Analytical interest is in one parameter: the probability of ‘success’. These terms are conventional — a ‘success’ can be a bad event.
What if you had many discrete outcomes? In the Savanta ComRes survey, there are seven response choices. Football teams can win, lose, or draw. For elections, there can be several political candidates to choose from. That leads to a general form of our Binomial distribution: the Multinomial distribution.
