An illustration of different ages of a woman.

According to an SSA study, around 39% of 65-year-old women will live to at least 90. An insurance company has a part of its portfolio made of life insurance policies sold to 65-year-old women. Based on the SSA study the insurance company expects that 61% of 65-year-old women will pass away before reaching the age of 90. A simple random sample of 450 is taken. Use the normal approximation along with the continuity correction factor to find the probability that at least 276 women in the sample will pass away before reaching the age of 90.
  • Let X\displaystyle {X} be the number of females out of 450 that pass away before reaching the age of 90. Describe the distribution of X\displaystyle {X} and its parameters:

X∼\displaystyle {X}\sim  ( n=\displaystyle {n}= , p=\displaystyle {p}= )

  • Use the random variable notation to symbolically express the probability that at least 276 females out of 450 pass away before reaching the age of 90:

  • Let Y\displaystyle {Y} be a normal variable that will be used to approximate the probability in question. Find the parameters of Y\displaystyle {Y} (round the answers to 2 decimal places):

 Y∼\displaystyle {Y}\sim ( μ=\displaystyle \mu= , σ=\displaystyle \sigma= )

  • Use the random variable notation to symbolically express the approximate probability that at least 276 females out of 450 pass away before reaching the age of 90:

  • Use the correction for continuity:

  • Find the probability (round the answer to 4 decimal places):

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