A 2013 article in the American Journal of Public Healthpresents the results of study of the relationship between social isolation and mortality.This analysis is based on datacollected in the Third National Health and NutritionalExamination Survey (NHANES), and the National Death Index.The following snippetcomes from the article abstract:

The authors chose to present the results of their analyses separately for females and males.This homework exercise will use only the results for females.

The following Kaplan-Meier curve shows the time to death after participation in NHANES 3, when the factors included in computing the social isolation index were measured.The curves are presented separately for the four social index score groups.

The authors report that "Among women, social isolation was associated with mortality (log rank p <0.001)"

- What is the null hypothesis for the log-rank test performed by the authors?

a. All four survival curves are equivalent at the population level.

b. At lease two of the four curves differ at the population level.