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This paper presents the results of an online survey experiment with 746 participants that investigated whether social norms influence people's choices about using technologies that can infer information they might not want to disclose. The results show both correlational and causal evidence that empirical expectations (beliefs about what others do) and normative expectations (beliefs about what others believe) influence choices to use mobile devices in ways that generate data that could be used to make sensitive inferences. However, participants also reported concern about data privacy, and lower behavioral intentions for vignettes involving more invasive inferences. Pluralistic ignorance is a phenomenon where individuals behave in ways they privately disagree with, because they see others around them behaving the same way and assume this is evidence most people approve of the behavior. These results are consistent with the existence of pluralistic ignorance related to data privacy, and suggest that interventions focused on transparency about data practices are not enough to encourage people to make different privacy choices.