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The Bradley effect concerns the discrepancy between opinion polls and actual election outcomes that emerges when candidates do not exhibit ideological, sexual or racial mainstream features. This effect was first observed during the 1982 election for the Governor of California that resulted in a significant loss for the black Democratic candidate, Tom Bradley, despite him being ahead in polls. It has been argued that poll respondents may tend to mask their true political preference in favour of what is generally considered more socially acceptable. We propose an exactly solvable statistical mechanical model, which allows for a quantitative exploration of this phenomenon. The model includes three main ingredients: (i) the tendency of individuals to align their real preference to the declared (public) opinions of others, (ii) a term accounting for an individual integrity factor, which induces voters to be consistent between their public and private opinions, and (iii) a self-monitoring component, which tunes the strength by which an individual is affected by and wishes to publicly conform to social norms. We observe the emergence of order in a wide range of parameters of the model and discuss the effect of noise on the correlation between hidden and public preference. Finally, we present an application of this model to the prediction of election outcomes.