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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?

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  • $\begingroup$ Is this a self-study question, like homework or something? If so, please add the "self-study" tag. $\endgroup$ – generic_user Sep 21 '18 at 15:18
  • $\begingroup$ @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. $\endgroup$ – Hydrochoerus Hydrochaeris Sep 22 '18 at 11:00
  • $\begingroup$ @HydrochoerusHydrochaeris were you able to solve your problem? Would be great if you can share your solution $\endgroup$ – info_seeker Oct 29 '18 at 17:51
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    $\begingroup$ Unfortunately @info_seeker, I did not manage to solve my problem and became only disillusioned with StackExchange as a source of knowledge. $\endgroup$ – Hydrochoerus Hydrochaeris Nov 11 '18 at 13:05
  • $\begingroup$ I am sorry for that. I feel there is a shortage of survey statisticians here. Perhaps reddit may help instead. $\endgroup$ – info_seeker Nov 11 '18 at 13:18

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