7. A factory produces individual cups of fruit salad. Each cup is lled with 15 pieces of fruit, taken randomly from a very large tank containing 20% of cherry pieces. A fruit cup is judged acceptable if it contains at least 2 pieces of cherry. (a) (2 points) Find the probability that a random fruit cup is acceptable. (b) (2 points) A bulk package contains 48 fruit cups. Find the probability that such a package contains exactly 40 acceptable fruit cups. 4 10. (4 points) In the United States, 0.6% of the population is allergic to peanuts. A random sample of 2500 people will be selected. We wish to calculate the probability that at least 10, but at most 20 people are allergic to peanuts in this sample. Use a normal approximation to nd this probability, and verify that it is valid to do so. l3. (3 points) Suppose the average weight of airline passengers is 65 kg with a standard deviation of 10 kg. A random sample of 81 people board a plane. If the plane only allows a maximum total weight of 5500 kg for its passengers, what. is the probability that this limit. will be exceeded? Carefully justify any result or formula that you are using. 9. According to the data obtained by the National Safety Council in the United States, 26% of all car crashes in 2014 were caused by using a cellphone while driving. (a) (2 points) What is the probability that, for the next 20 car crashes, exactly 5 of them will be caused by using a cellphone? (b) (4 points) What is the probability that, for the next 1000 car crashes in the United States, at least 250 and at most 300 of them will be caused by using a cellphone? Use a normal approximation, and verify rst that it is valid to use such an approximation. 10. Suppose that the lifespan of a car battery follows a normal distribution with an average of 5 years, and a standard deviation of 1.2 years. (a) (2 points) Find the probability that a given car battery will last more than 6 years. (b) (2 points) If we take a random sample of 10 car batteries, what is the probability that the average lifespan of these batteries will exceed 5 years? (c) (2 points) Again using a sample of 10 batteries, what is the probability that exactly 5 of them will last more than 6 years? \f8. If X is a Poisson random variable with 2=6, find E(y) where y=3x+7. 9. If X is a Binomial random variable with n=6 and p=0.5, find the probability that x takes on a value within one standard deviation of the mean