Question
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 66 years and a standard deviation of 4 years. What proportion of the plan recipients would receive payments beyond age 75?
Answers
GPT-4o mini
To determine the proportion of plan recipients who would receive payments beyond age 75 (i.e., the proportion who live longer than 75 years), we first need to standardize the age of 75 using the properties of the normal distribution. We will use the z-score formula:
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \(X\) is the value we are standardizing (in this case, 75 years),
- \(\mu\) is the mean of the distribution (66 years),
- \(\sigma\) is the standard deviation of the distribution (4 years).
Plugging in the values:
\[
z = \frac{75 - 66}{4} = \frac{9}{4} = 2.25
\]
Next, we will look up this z-score in the standard normal distribution table or use a calculator to find the cumulative probability. The cumulative probability tells us the proportion of plan recipients that live **up to** age 75.
Finding the probability for \(z = 2.25\):
Using a standard normal distribution table, we find:
\[
P(Z \leq 2.25) \approx 0.9878
\]
This means that approximately 98.78% of participants are expected to die before reaching age 75.
To find the proportion of participants who receive payments beyond age 75, we need to calculate the complement of this probability:
\[
P(Z > 2.25) = 1 - P(Z \leq 2.25) \approx 1 - 0.9878 = 0.0122
\]
Thus, the proportion of plan recipients who would receive payments beyond age 75 is approximately **0.0122**, or **1.22%**.
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \(X\) is the value we are standardizing (in this case, 75 years),
- \(\mu\) is the mean of the distribution (66 years),
- \(\sigma\) is the standard deviation of the distribution (4 years).
Plugging in the values:
\[
z = \frac{75 - 66}{4} = \frac{9}{4} = 2.25
\]
Next, we will look up this z-score in the standard normal distribution table or use a calculator to find the cumulative probability. The cumulative probability tells us the proportion of plan recipients that live **up to** age 75.
Finding the probability for \(z = 2.25\):
Using a standard normal distribution table, we find:
\[
P(Z \leq 2.25) \approx 0.9878
\]
This means that approximately 98.78% of participants are expected to die before reaching age 75.
To find the proportion of participants who receive payments beyond age 75, we need to calculate the complement of this probability:
\[
P(Z > 2.25) = 1 - P(Z \leq 2.25) \approx 1 - 0.9878 = 0.0122
\]
Thus, the proportion of plan recipients who would receive payments beyond age 75 is approximately **0.0122**, or **1.22%**.