# Statistical significance of percentage difference for a poll question with multiple options?

The question is: how to calculate the statistical significance of difference between percentages in the case described below?

Suppose we have a poll question with $$n$$ answer options $$A_1, …, A_n$$, where a respondent can choose up to $$k$$ options simultaneously ($$1 \leq k \leq n$$). The option $$A_x$$ scored $$x$$% of votes, and the option $$A_y$$ scored $$y$$%. (In general, any respondent might choose both $$A_x$$ and $$A_y$$.)

How to calculate whether the difference between $$x$$ and $$y$$ is significant at a given significance level (say, .95)? I asked a question similar to this one formerly, but only the case $$k = 1$$ was considered then. Now I look for a test for more general case.

UPD. Here is a live example from my current research:

247 respondents have answered the question about their preferred kind of investments. 170 of them chose hospitality property, and 124 chose retail property. There were 3 more options in the question (5 total), and each respondent was able to choose up to 2 options. So is the difference between 124 and 170 statistically significant enough to say that hospitality property is more popular asset than retail? How to find out?

• Is this a self-study question, like homework or something? If so, please add the "self-study" tag. – generic_user Sep 21 '18 at 15:18
• @generic_user , this is part of handling the results of a large survey conducted by a real estate company that I am working for. The summary of our results will be published in the WSJ and other business media, as our previous surveys were. – Hydrochoerus Hydrochaeris Sep 22 '18 at 11:00
• @HydrochoerusHydrochaeris were you able to solve your problem? Would be great if you can share your solution – info_seeker Oct 29 '18 at 17:51
• Unfortunately @info_seeker, I did not manage to solve my problem and became only disillusioned with StackExchange as a source of knowledge. – Hydrochoerus Hydrochaeris Nov 11 '18 at 13:05
• I am sorry for that. I feel there is a shortage of survey statisticians here. Perhaps reddit may help instead. – info_seeker Nov 11 '18 at 13:18