In a mortality study, the following grouped death data were collected from 100 lives, all studied beginning at age 40 .

Age last birthday at death	Number of deaths
[math]40-49[/math]	10
[math]50-59[/math]	14
[math]60-69[/math]	16
[math]70-79[/math]	20
80 and higher	40

There were no terminations other than death.

Calculate [math]\hat{S}_{40}(32)[/math] using the ogive empirical distribution function.

• 0.44
• 0.48
• 0.52
• 0.56
• 0.60

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

Answer: D

[math]\hat{S}(30)=\frac{100-10-14-16}{100}=0.60, \quad \hat{S}(40)=\frac{100-10-14-16-20}{100}=0.40[/math]

Use linear interpolation to find [math]\hat{S}(32)[/math]

[math]\hat{S}(32)=\left(\frac{40-32}{40-30}\right) \hat{S}(30)+\left(\frac{32-30}{40-30}\right) \hat{S}(40)+=0.8(0.60)+0.2(0.40)=0.56[/math]

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