Testing statistical significance of male vs female enrolment preference percentages between two different fields of education The problem I have is as follows.  I have enrolment data describing the number of male and female preferences for two broad fields of education: i) Information Technology and ii) Engineering.  The data reflects "first preferences", that is, the number of people who wrote down on paper that their most preferred degree was either one in the field of Information Technology or the field of Engineering.  The data does not reflect actual enrolments.  The numbers are as follows:
Information Technology:
Female: 266  (~13%)
Male: 1783  (~87%)
Engineering:
Female: 684  (~12.5%)
Male: 4773  (~87.5%)
The question I am trying to answer is: is there a statistically significant difference between the gender ratios of Engineering versus Information Technology, and at what level of significance? 
I apologize if this is a noob question.  I only have relatively basic background knowledge in statistics and tests of significance and this problem didn't seem to fit any of the methods I am familiar with.  I'm hoping someone might be able to point me in the right direction as to which statistical test(s) would be appropriate given the dataset and any caveats I should be aware of.
 A: You are looking to test for a "difference between differences," as James Jaccard would say.  "Does the gender difference come out differently depending on whether the field is IT or Engineering?" (I would't pursue the question by looking at ratios per se.)  Such questions are typically addressed by testing for statistical interactions in a regression or anova model.  In this case you have a binary outcome--enrolled or didn't enroll--so logistic regression would be the natural choice.
With that said, it's hard to imagine too many people caring whether the gender-by-field interaction is statistically significant when it looks so insignificant in practical terms.  
A: Your problem is very much similar to one of the most well known real life examples of Simpson's paradox occurred when the University of California, Berkeley was sued for bias against women who had applied for admission to graduate schools there. The admission figures for the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance.
But when examining the individual departments, it appeared that no department was significantly biased against women. In fact, most departments had a "small but statistically significant bias in favor of women."
I think you would like to see the research paper by Bickel, et al. on this topic. You will see that he has used the chi-square test just like Glen_b has suggested you. I hope this helps! 
