To find the proportion of plan recipients who die before reaching the standard retirement age of 65, we calculate the z-score and then use the standard normal distribution table.

Step 1: Z-Score Calculation

The z-score is given by the formula:

$z = \frac{x - \mu}{\sigma}$

Where:

• $x = 65$ (the age of interest),
• $\mu = 67$ (mean lifetime of participants),
• $\sigma = 2.6$ (standard deviation of lifetimes).

Substituting the values:

$z = \frac{65 - 67}{2.6} = \frac{-2}{2.6} \approx -0.769$

Step 2: Find the Proportion Below z = -0.769

Using a standard normal distribution table or a calculator, the cumulative probability for $z = -0.769$ is approximately:

$P(Z < -0.769) \approx 0.2206$

Interpretation

Approximately 22.06% of the plan recipients are expected to die before reaching the standard retirement age of 65.

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Related Questions:

1. What is the proportion of recipients expected to live beyond 70 years?
2. What is the probability that a recipient lives between 65 and 70 years?
3. How does the mean and standard deviation affect the probability calculation?
4. If the standard deviation were increased to 3 years, how would the proportion below 65 years change?
5. What proportion of participants live at least one standard deviation above the mean?

Tip:

When working with normal distributions, always visualize the problem to better understand the cumulative areas under the curve.