According to a study in The New England Journal of Medicine, 202 of a sample of 5,990 middle-aged men had developed diabetes. It also found that men who were very active (burning about 3,500 calories daily) were half as likely to develop diabetes compared with men who were sedentary. Assume that one-fourth of all middle-aged men are very active, and the rest are classified as sedentary. What is the probability that a middle-aged man with diabetes is very active Please, round your answer to two decimal places.

Question

According to a study in The New England Journal of Medicine, 202 of a sample of 5,990 middle-aged men had developed diabetes. It also found that men who were very active (burning about 3,500 calories daily) were half as likely to develop diabetes compared with men who were sedentary. Assume that one-fourth of all middle-aged men are very active, and the rest are classified as sedentary. What is the probability that a middle-aged man with diabetes is very active  Please, round your answer to two decimal places.

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0.14Given that 202 out of 5,990 men developed diabetes, and assuming one-fourth of all middle-aged men are very active, we can calculate the probability that a middle-aged man with diabetes is very active using Bayes' theorem. Let A be the event that a man is very active, and B be the event that a man has diabetes. We need to find P(A|B), the probability a man is very active given that he has diabetes.First, calculate the probabilities:- P(A) = 1/4 (probability a man is very active)- P(A') = 3/4 (probability a man is sedentary)- P(B|A) = 1/2 * P(B|A') (very active men are half as likely to develop diabetes as sedentary men)- P(B) = 202/5990 (overall probability of developing diabetes)Using Bayes' theorem:$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$Given that very active men are half as likely to develop diabetes, let's assume sedentary men are twice as likely to be in the diabetes group. If 1/4 of men are very active, then in a simplified model, for every 4 men, 1 is very active and 3 are sedentary. If sedentary men are twice as likely to develop diabetes, then for every 3 cases of diabetes in sedentary men, there would be 1 case in very active men, assuming the incidence in the very active group is half that of the sedentary group.However, to accurately calculate P(A|B) without specific numbers for P(B|A) and P(B|A'), we use the given information to infer proportions. The exact calculation requires knowing the actual rates of diabetes in very active and sedentary men, which we simplify based on the given information.Given the lack of specific incidence rates, we can infer that the probability a man with diabetes is very active is significantly lower than the probability he is sedentary, due to the protective effect of being very active against diabetes. The correct answer, based on the information provided and the need to apply Bayes' theorem correctly with detailed data, is a simplified estimation.The calculation simplifies to understanding that very active men are less likely to develop diabetes, and since they represent a smaller portion of the population, the probability that a man with diabetes is very active is relatively low. The correct choice, based on the information and typical application of Bayes' theorem to such problems, is closest to the provided options without detailed calculation.

According to a Study in the New England Journal of Medicine