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8.1 Prior, Likelihood, and Posterior for Two Proportions 157 circumstances of war but who never forget each other. The other random half of your participants is shown a light comedy about high school pranks. Imme- diately after seeing the film, all participants are given some cognitive tasks, including an arithmetic problem involving long division. Of the 40 people who saw the war movie, 32 correctly solved the long division problem. Of the 40 people who saw the comedy, 27 correctly solved the long division problem. Did the induced mood actually affect cognitive performance? In other words, based on the observed difference in proportions, 32/40 versus 27/40, what underlying difference is actually credible? To talk about the problem more generally and with mathematical precision, we need to define some notation. We'll use the same sort of notation that we've used for previous chapters, but we'll add subscripts to indicate which of the two groups is being referred to. Thus, the hypothesized proportion of "heads" in group j (where j = 1 or j = 2) is denoted j , and the actual number of heads observed in a sample of N j "flips" is z j , and the i th individual flip in group j is denoted y ji . Throughout this chapter we assume that the data from the two groups are inde-