A club is established with 2000 members, 1000 of exact age 35 and 1000 of exact age 45 . You are given:

(i) Mortality follows the Standard Ultimate Life Table

(ii) Future lifetimes are independent

(iii) [math] N[/math] is the random variable for the number of members still alive 40 years after the club is established

Using the normal approximation, without the continuity correction, calculate the smallest [math]n[/math] such that [math]\operatorname{Pr}(N \geq n) \leq 0.05[/math].

• 1500
• 1505
• 1510
• 1515
• 1520

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Answer: B

[math]E(N)=1000\left({ }_{40} p_{35}+{ }_{40} p_{45}\right)=1000\left(\frac{85,203.5}{99,556.7}+\frac{61,184.9}{99,033.9}\right)=1473.65[/math]

[math]\operatorname{Var}(N)=1000_{40} p_{35}\left(1-{ }_{40} p_{35}\right)+1000_{40} p_{45}\left(1-{ }_{40} p_{45}\right)=359.50[/math]

Since [math]1473.65+1.645 \sqrt{359.50}=1504.84[/math]

[math]N=1505[/math]

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Copyright 2024. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.