# what is the probability that the discrepancy between proportions being promoted would be at least this different by chance alone

A police officer in an urban police department alleges that minorities are being discriminated against in promotion decisions. The difference in promotion rates in 1984 is offered as evidence. In that year, among those eligible for promotion to the rank of sergeant, 20 officers, including 7 members of minority groups, passed an objective exam to qualify them for consideration by the review board. The number of officers that can be promoted is determined by the number of vacancies at the higher rank, and in 1984, there were 10 vacancies at the rank of sergeant that needed to be filled. Eight of the 13 nonminority officers were promoted, for a promotion rate of 61.5%, whereas only 2 of the 7 minority officers were promoted, for a promotion rate of 28.5%. If one assumes that the decisions about whom to promote were made independently of minority status, what is the probability that the discrepancy between proportions being promoted would be at least this different by chance alone, given the total number of officers under consideration and the total number of promotions possible?

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