SOLUTION: 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 distributi

Question 1022427: 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. Find the age at which payments have ceased for approximately 86% of the plan participants.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
mean is 68.
standard deviation is 3.5.

you need to find the z-score that gives you a percentage to the left of that z-score of .85.

that would be a z-factor of 1.03643338, derived using my calculator.

we can round that off to 1.04.

if you look that up in the z-score table, you will see that the area of the distribution curve to the left of a z-score of 1.04 is equal to .8508.

not right on, but pretty close.

now we need to find the age that the z-score of 1.04 represents.

the formula for z-score is z = (x-m)/s

z is the z-score.
x is the raw score used to find the z-score.
m is the mean used to find the z-score.
s is the standard deviation used to find the z-score.

in this problem, the formula becomes 1.04 = (x - 68) / 3.5.

the z-score is 1.04.
the raw score is what we want to find.
the mean life expectancy is equal to 68 years of age.
the standard deviation is 3.5 years of age.

we'll use this formula to find the value of x.

solve for x to get x = 1.04 * 3.5 + 68.

this results in x = 71.64.

given that the life expectancy is a normal distribution, and given that the average life expectancy is 68 years of age and that the standard deviation is 3.5 year, then the probability that a person will die by the age of 71.64 is 85%.

since the payments cease when the person is dead, then the payments will have ceased for approximately 85% of the plan recipients after they have attained the age of 71.64 years.

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