Those are value judgements, they cannot be "wrong". You can judge them as wrong, but that's just one value judgement on top of another. There is no way to even hold a discussion between them on that ground.
> An indicator of the prevalence of these beliefs is the dominance of women in the nursing profession. This is hard, factual, statistical data that we can point to
This is hard, factual, statistical data that doesn't support your conclusion. Maybe women really do have a stronger preference for nursing jobs, or they are socialized that way, or those jobs fit their schedule better, or they like working with other women so they congregate in jobs already dominated by other women, or..., or some combination of the above.
Why take one piece of data and jump to the least charitable explanation? Sexism is just one of very many possible causes.
> Are blacks more likely to be criminals, or does the expectation that blacks become criminals increase the number of blacks being investigated for crime, leading to a higher rate of discovery?
Again, or both, or neither. Or both but maybe some other factor completely dwarfs those two you mention, rates of single motherhood come to mind.