Please show all correct work

Please show all correct work
BUSN 3400 – FINAL EXAM TAKE HOME WORKED PROBLEMS – SHOW CALCULATIONS IN ORDER TO GET FULL CREDIT. 1. A company that sells annuities must base the annual payout on the probability distribution of the length of life of the participants in the plan. Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years. What proportion of the plan recipients would receive payments? [8 points] (a) less than age 75? (b) more than 68 years old? (c) between 60 and 64 years old? (d) between 65 and 69 years old? 2. At a computer manufacturing company, the actual size of computer chips is normally distributed with a mean of 10 centimeter and a standard deviation of 2.1 centimeter. A random sample of 24 computer chips is taken. What is the probability that the sample mean will be between 10.95 and 11.05 centimeters? [3 points] 3. The amount of a time a bank teller spends with each customer has a population mean, of 3.10 minutes and standard deviation, of 0.40 minutes. If you select a random sample of 16 customers, [5 points] What is the probability that the mean time spent per customer is at least 3 minutes? There is an 85% chance that the sample mean is less than how many minutes? 4. The personnel director of a large corporation wants to study absenteeism among clerical workers at the corporation’s central office during the year. A random sample of 25 clerical workers reveals the following: [20 points] Absenteeism: = 9.7 days, S = 4.0 days. 12 clerical workers were absent more than 10 days. Set up a 95 % confidence interval estimate of the average number of absences for clerical workers last year. Set up a 95% confidence interval estimate of the population proportion of clerical workers absent more than 10 days last year. If the personnel director also wants to take a survey in a branch office, what sample size is needed if the director wants to be 95% confident of being correct to within  1.5 days and the population standard deviation is assumed to be 4.5 days? If the personnel director also wants to take a survey in a branch office, what sample size is needed if the director wants to be 90% confident of being correct to within  0.075 of the population proportion of workers who are absent more than 10 days if no previous estimate is available?