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From a survey I have answers towards a yes/no question and the survey-respondents are grouped into multiple groups. I now want to know if the proportion of yes-answers for one group differs from the whole survey sample (thus including the subgroup).

I would do this by calculating the standard error. I could do it using the following formula:

$\text{SE} = \sqrt{p(1-p)/n}$

where $p$ is the overall proportion of all yes and $n$ is the total number of answers via all groups.

In the comments to this question it is stated: ". You can test the difference in means between the subset and the overall group as long as you account for the covariance between $\bar{X}_d$ and $\bar{X}$ in your standard error calculation" (with $\bar{X}_d$ being the answers for the subgroup and $\bar{X}$ being the answers for the entire sample).

My question is how to incorporate the covariance into my standard error calculation.

Glen_b
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  • https://en.wikipedia.org/wiki/Variance#Basic_properties ... so $\text{var}(X-Y) = ...$ ...? – Glen_b Jun 21 '17 at 08:59
  • hi Glen, can you give it some more words. right now that doesn't help me much. thanks! – Jan Jun 21 '17 at 09:26
  • Sorry, I misread one sentence in your question, which led me to think this was a homework assignment. How does this issue arise? In particular, note that the subgroup doesn't differ from itself, so it can *only* differ from the whole if it differs from its complement. So you answer the subgroup vs all question by answering the subgroup vs everyone else question - it's logically identical but simpler. In short, what *prevents* you from comparing the subgroup to everyone but the subgroup? Basically, only if you can't make that comparison for some reason would the longer calculation be worthwhile – Glen_b Jun 21 '17 at 09:29
  • It is Phd-thesis related, so the problem somewhat arose naturally and without external forces ;) I want to see if there are differences between the total-market and a specific market segment. So I think both ways subgroup vs. a) total-market and b) total-market minus subgroup are possible. For me version a) would be more logical as it would fit better in my analyses asking "is this segment different from the total-market" – Jan Jun 21 '17 at 10:29
  • Also see https://stats.stackexchange.com/questions/83225/testing-whether-the-mean-of-a-group-is-different-from-the-mean-of-the-entire-sam and https://stats.stackexchange.com/questions/30562/how-to-test-whether-subgroup-mean-differs-from-overall-group-that-includes-the – Glen_b Jul 10 '17 at 11:12
  • thanks, Glen! juts flagged my question as duplicate of the first question you mentioned :) – Jan Jul 10 '17 at 13:54

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