I'm having a hard to deciding which of a two-sample t-test and a chi-squared test is more appropriate for these sorts of questions. Examples would be:
70 out of 120 americans like hotdogs. 50 out of 95 europeans like hotdogs. Do the two groups differ in their hotdog opinions?
1000 each of caucasians, african-americans, and asians were asked if they preferred red, white, or yellow onions (counts provided in a table). Do respondents from different ethnicities differ in their responses?
I realize that those questions are worded slightly differently – I think it may be the wording that differentiates them? In either case, i'm confused about which of the two tests is more appropriate.
You could use chi-squared tests to answer those questions. (Note that they're all about comparing proportions/counts – that's usually a clue.)
An alternative (particularly handy if you want a one-tailed test) would be a two-sample proportions test. In the two-tailed case they will give the same p-value (as long as they both do or both don't do the continuity correction).
(Let's say you plan to do a t-test. Once you figure out how to get a sample standard deviation, the resulting statistic doesn't have a t-distribution. The numerator isn't normal, the denominator isn't the square root of a scaled chi-squared and the numerator and denominator aren't independent.)
Google things like comparing two proportions and comparison of proportions and note how they're done. They're not often done with t-tests.
You can make a t-test work (by using 0s and 1s as the data), and it's generally going to work reasonably well, but as far as I know we have no solid basis for arguing that it's going to be better than doing the usual normal approximation.