Questionnaires are by far the most common tool for measuring noncognitive constructs in psychology and educational sciences. Response bias may pose an additional source of variation between respondents that threatens validity of conclusions drawn from questionnaire data. We present a mixture modeling approach that leverages response time data from computer-administered questionnaires for the joint identification and modeling of two commonly encountered response bias that, so far, have only been modeled separately--careless and insufficient effort responding and response styles (RS) in attentive answering. Using empirical data from the Programme for International Student Assessment 2015 background questionnaire and the case of extreme RS as an example, we illustrate how the proposed approach supports gaining a more nuanced understanding of response behavior as well as how neglecting either type of response bias may impact conclusions on respondents' content trait levels as well as on their displayed response behavior. We further contrast the proposed approach against a more heuristic two-step procedure that first eliminates presumed careless respondents from the data and subsequently applies model-based approaches accommodating RS. To investigate the trustworthiness of results obtained in the empirical application, we conduct a parameter recovery study.