In a study of 1,000 people with a particular illness, 200 died within one year of diagnosis. Calculate a 95% (linear) confidence interval for the one-year empirical survival function.

• (0.745,0.855)
• (0.755,0.845)
• (0.765,0.835)
• (0.775,0.825)
• (0.785,0.815)

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

Answer: D

[math]\hat{S}(1)=0.8[/math]

[math]\operatorname{Var}[S(1)]=\frac{S(1)(1-S(1)}{n} \approx \frac{(0.8)(0.2)}{1000}=0.01265^{2}[/math]

[math]\Rightarrow 95 \% \mathrm{CI}[/math] is approximately [math](0.8 \pm 1.96(0.01265))=(0.775,0.825)[/math]

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