Suppose I am taking a survey sample of how many people own a cat. Obviously there is only two categorical outcomes: yes or no. However suppose I want to expand my survey and determine what cat names are most popular. Again this is categorical with 'no cat' being one option but then many other possible options for names.
My question is therefore how do you calculate the standard error in cases like this, and how do you determine an appropriate sample size?
For instance in the first example suppose my sample size if effectively infinite (many millions of people, a whole country's population). Since I'm just looking at a yes/no answer I probably don't need that many random samples to get a fairly accurate answer. However in the second example (cat names) because there are so many more possibilities I expect my minimum sample size would have to be much larger than the simple yes/no survey, but how do I determine how large it would need to be in order to be representitive of the larger population?