n a survey, 500 randomly selected adults who drink coffee were askedwhether they usually drink coffee with or without sugar. Of these 500adults, 240 are men, and 175 drink coffee without sugar. Of the 240men, 84 drink coffee without sugar.40. ***Which of the following proves that the events drinking coffeewithout sugar and man are independent?I. P(drinks coffee with sugar) = ଷଶହହ = 0.65II. P(drinks coffee without sugar) = ଵହହ = 0.35III. P(man | drinks coffee without sugar) = ଼ ସଵହ = 0.48IV. P(drinks coffee without sugar | man) = ଼ ସଶସ = 0.35A. I and IIB. I and IIIC. II and IIID. II and IV41. Which of the following events are mutually exclusive?A. Woman and drinking coffee with sugarB. Man and drinking coffee without sugarC. Woman and drinking coffee without sugarD. Drinking coffee with sugar and drinking coffee without sugar42. A pregnancy test reporting that a woman is pregnant eventhough she is not, is an example of aA. Type I errorB. Type II errorC. Type III errorD. Type IV error
Question
n a survey, 500 randomly selected adults who drink coffee were askedwhether they usually drink coffee with or without sugar. Of these 500adults, 240 are men, and 175 drink coffee without sugar. Of the 240men, 84 drink coffee without sugar.40. ***Which of the following proves that the events drinking coffeewithout sugar and man are independent?I. P(drinks coffee with sugar) = ଷଶହହ = 0.65II. P(drinks coffee without sugar) = ଵହହ = 0.35III. P(man | drinks coffee without sugar) = ଼ ସଵହ = 0.48IV. P(drinks coffee without sugar | man) = ଼ ସଶସ = 0.35A. I and IIB. I and IIIC. II and IIID. II and IV41. Which of the following events are mutually exclusive?A. Woman and drinking coffee with sugarB. Man and drinking coffee without sugarC. Woman and drinking coffee without sugarD. Drinking coffee with sugar and drinking coffee without sugar42. A pregnancy test reporting that a woman is pregnant eventhough she is not, is an example of aA. Type I errorB. Type II errorC. Type III errorD. Type IV error
Solution
- Two events are independent if the probability of one event occurring does not affect the probability of the other event occurring. In this case, the events are "drinking coffee without sugar" and "being a man". These events would be independent if P(man and drinks coffee without sugar) = P(man) * P(drinks coffee without sugar).
From the information given, we can calculate these probabilities as follows:
P(man) = 240/500 = 0.48 P(drinks coffee without sugar) = 175/500 = 0.35 P(man and drinks coffee without sugar) = 84/500 = 0.168
If we multiply P(man) and P(drinks coffee without sugar), we get 0.48 * 0.35 = 0.168. Since this is equal to P(man and drinks coffee without sugar), the events are independent. Therefore, the answer is D. II and IV.
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Mutually exclusive events cannot occur at the same time. In this case, the only pair of events that cannot occur at the same time is "drinking coffee with sugar" and "drinking coffee without sugar". Therefore, the answer is D. Drinking coffee with sugar and drinking coffee without sugar.
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A Type I error, also known as a false positive, occurs when a test incorrectly indicates the presence of a condition. In this case, the pregnancy test is indicating that the woman is pregnant when she is not. Therefore, the answer is A. Type I error.
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