ABSTRACT
For example, “The probability is at least 95% that the fraction of people who prefer apple pie is between 54% and 60%” would bemore descriptive. Symbolically, let f be the true fractionof thepopulation and let f¯ = 0.574be the fraction in the poll. Then the claim is P (0.54 ≤ f ≤ 0.60 | f¯ = 0.574) ≥ 0.95. The interval [0.54,0.60] is called the confidence interval for f at the 95% level. (Sometimes it is called the confidence interval at the 5% level. Since no one is ever interested in intervals where the probability is less than 1/2, this does not give rise to ambiguity.)
The actual calculation is quite simple, but the justification is complicated. In fact, the same calculation can be justified in two quite different ways, one corresponding to the likelihood theory of probability and one corresponding to the frequentist theory. Since 99% of the time, all you need is to do the calculation (and 99% of the people who do these calculations neither understand the justification nor care about it), wewill begin here how to do the calculation, andwepoint out some important features of it. The justifications are discussed in optional Sections 11.3 and 11.4.
