Solve the following questions

B. Binomial Probability Distribution 1. A safety engineer claims that only 40% of all workers wear safety helmets when they eat lunch atthe workplace. Assuming that this claim is right, find the probability that 4 of 6 workers randomly chosenwill be wearing their helmets while having lunch at the workplace. 2. Suppose that for a very large shipment ofintegrated-circuit chips, the probability of failure for any one chip is 0.10. Assuming that the assumptions underlying the binomial distributions are met, find the probability that at most 3 chips fail in a random sample of 20. 3. Assuming that 6 in 10 automobile accidents are due mainly to a speed violation, find the probability that among 8 automobile accidents, 6 will be due mainly to a speed violation. 4. If the probability that a fluorescent light has auseful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights (a) exactly 18 will have a useful life of at least 800 hours; (b) at least 15 will have a useful life of at least 800 hours; (c) at least 2 will not have a useful life of at least 800 hours. 5. A traffic control engineer reports that 75% of the vehicles passing through a checkpoint are from within the state. What is the probability that fewer than 4 of the next 9 vehicles are from out of state? C. Normal Probability Distribution 1. The tensile strength of a certain metal component is normally distributed with a mean of 10,000 kilograms per square centimeter and a standard deviation of 100 kilograms per square centimeter. Measurements are recorded to the nearest 50 kilograms per square centimeter. (a) What proportion of these components exceeds 10, 150 kilograms per square centimeter in tensile strength? (b) If specifications require that all components have tensile strength between 9800 and 10,200 kilograms per square centimeter inclusive, what proportion of pieces would we expect to scrap? 2. Given a normal distribution with / = 30 and o =6, find (a) the normal curve area to the right of x = 17; (b) the normal curve area to the left of x = 22; (c) the normal curve area between x = 32 and x = 41; (d) the value of x that has 80% of the normal curvearea to the left; (e) the two values of x that contain the middle 75% ofthe normal curve area. 3. Research scientist reports that mice will live an average of 40 months when their diets are sharply restricted and then enriched with vitamins and proteins. Assuming that the lifetimes of such mice are normallydistributed with a standard deviation of 6.3 months, find the probability that a given mouse will live (a) more than 32 months; (b) less than 28 months; (c) between 37 and 49 months. 4. A lawyer commutes daily from his suburban home to his midtown office. The average time for aone-way trip is 24 minutes, with a standard deviationof 3.8 minutes. Assume the distribution of trip times to be normally distributed. (a) What is the probability that a trip will take at least 1/2 hour? (b) If the office opens at 9:00 A.M. and the lawyer leaveshis house at 8:45 A.M. daily, what percentage of thetime is he late for work? (c) If he leaves the house at 8:35 A.M. and coffee isserved at the office from 8:50 A.M. until 9:00 A.M., what is the probability that he misses coffee? (d) Find the length of time above which we find the slowest 15% of the trips. (e) Find the probability that 2 of the next 3 trips willtake at least 1/2 hour. 5. The IQs of 600 applicants to a certain college are approximately normally distributed with a meanof 115 and a standard deviation of 12. If the college requires an IQ of at least 95, how many of these students will be rejected on this basis of IQ, regardless oftheir other qualifications? Note that IQs are recorded to the nearest integers