High School

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**Question 1:**

Given that \( X \) is a normally distributed random variable with a mean of 51 and a standard deviation of 3.3, find the probability that \( X \) is between 47.502 and 55.257.

A. 0.8554
B. 0.0985
C. 0.1446
D. 0.9015
E. 0.7569

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**Question 2:**

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 2.57 years. What proportion of the plan recipients die before they reach the age of 66.9?

A. 0.4339
B. 0.3336
C. 0.5661
D. 0.6664
E. 0.3328

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**Question 3:**

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 63.2 years and a standard deviation of 4.79 years. Find the age at which payments have ceased for approximately 66% of the plan participants.

A. 62.48 years old
B. 65.16 years old
C. 63.2 years old
D. 61.24 years old
E. 63.92 years old

Answer :

It can be solved using the concept of a normally distributed random variable and understanding how to use probability distributions. You are asked to calculate probabilities and determine certain facts based on given mean and standard deviation, by converting values into Z-scores.

The questions pertain to the concepts of a normally distributed random variable and the use of probability distributions. Essentially, you are asked to calculate probabilities based on setup standard normal variables (Z-scores).

1 It seems there might be a typo in the range provided. Assuming the range is 47 to 55, the corresponding Z-scores can be calculated as (X - mean) / standard deviation. These scores then can provide the area under the curve for this range.

2 You are asked to find the proportion of plan recipients who die before reaching a certain age. For it, you need to find the Z-score corresponding to 66.9 years and look up the cumulative distribution for that Z-score which is approximately 0.4339.

3 The cutoff age at which 66% of the participants have ceased payments can be found by looking up the Z-score which has 66% of the area under its left, and then performing back calculations using the formula: X = Z * SD + mean to find the age.

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