Initially, 80 lives are included in an observation of survival times following a specific medical treatment. You are given excerpted information from the study data in the table below.

[math]j[/math]	[math]t_{(j)}[/math]	Deaths at [math]t_{(j)}[/math]	Exits (Other than deaths) in [math](t_{(j)},t_{(j+1)}][/math]	Entrants in [math](t_{(j)},t_{(j+1)}][/math]
0		20	4
1	0.5	1	2	3
2	1.6	1	6	0
3	1.9	1	8	0
4	2.5	1	10	0

Calculate the Kaplan-Meier estimate of [math]S(2)[/math].

• 0.931
• 0.952
• 0.960
• 0.969
• 0.972

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

Answer: B

[math]r_{1}=80-20+4=64 ; \quad r_{2}=63-2+3=64 ; \quad r_{3}=63-6=57[/math]

[math]\hat{S}(2)=\hat{S}\left(t_{(3)}\right)=\frac{63}{64} \times \frac{63}{64} \times \frac{56}{57}=0.951994[/math]