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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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