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. a) What proportion of the plan recipients would receive payments beyond age 75? (Explain/show how you obtain your answer.) b) What proportion of the plan recipients die before they reach the standard retirement age of 65? (Explain/show how you obtain your answer.) c) Find the age at which payments have ceased for approximately 86% of the plan participants. (Explain/show how you obtain your answer.) 2. Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in 2009. Suppose a sample of 100 major league players was taken. a) What was the standard error for the sample mean salary? (Explain/show how you obtain your answer.) b) Find the approximate probability that the mean salary of the 100 players exceeded $3.5 million. (Explain/show how you obtain your answer.) c) Find the approximate probability that the mean salary of the 100 players exceeded $4.0 million. (Explain/show how you obtain your answer.) d) Find the approximate probability that the mean salary of the 100 players was no more than $3.0 million. (Explain/show how you obtain your answer.) e) Find the approximate probability that the mean salary of the 100 players was less than $2.5 million. (Explain/show how you obtain your answer.) 3. At a computer manufacturing company, the actual size of computer chips is normally distributed with a mean of 1 centimeter and a standard deviation of 0.1 centimeter. A random sample of 12 computer chips is taken. a) What is the standard error for the sample mean? (Explain/show how you obtain your answer.) b) What is the probability that the sample mean will be between 0.99 and 1.01 centimeters? (Explain/show how you obtain your answer.) c) What is the probability that the sample mean will be below 0.95 centimeters? (Explain/show how you obtain your answer.) d) Above what value do 2.5% of the sample means fall? (Explain/show how you obtain your answer.)