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 (Show your workings clearly).a) What proportion of the plan recipients would receive payments beyond age 75?b) What proportion of the plan recipients die before they reach the standard retirement age of 65?c) Find the age at which payments have ceased for approximately 86% of the plan participants.

Answer:answer: At age 71.78 86% of the payments of the plan participants will be ceased Step-by-step explanation:Let X be the age at which a person dies. We know that X has a normal distribution with mean 68 and standard deviation 3.5. REcall that the expression P(X<=x) means the probability that a person dies before the age x. In order for us to do the calculations, we consider the random variable Z = (X-68)/3.5. The variable Z will be normally distributed with mean 0 and standard deviation 1 (i.e will be a standard normal distribution). Based on that, we would like to calculate the Z value for which we have 86% of the curve on the  left of the value. Using a standard normal distribution table, we can find the  z value corresponding to the 86% of the curve, which is aproximately 1.08. Then, the age we are looking for is 1.08*3.5+68 = 71.78 years