In a frequently cited paper, Becker (1974) asked whether data from a “mortal” (= terminated) smallpox outbreak better fit a subcritical or semicritical Galton-Watson branching process. We revisit this question using standard hypothesis-testing methods and reach a somewhat di↵erent conclusion than he did. For a non-terminated process, the related problem of predicting extinction or explosion also can be formulated in a testing framework to which standard Neyman-Pearson theory applies.