I have the following 2x2 table                  NO     YES                 9690   5    LOW                 26354  39   HIGH   I want to   plot the proportions and their exact confidence intervals in two bar plots test if the proportion YES in LOW differs significantly from the proportion YES in high (2 sided test.   1.  I used the binconf funtion, Hmisc package in R to compute the exact condidence intervals  binconf(5,9690+5,alpha=0.05)  PointEstimate of proportion: 0.0005157298  CI.LOWER BOUND: 0.0002203084  CI.UPPER BOUND: 0.001206817  binconf(39,26354+39,alpha=0.05)  PointEstimate of proportion: 0.001477665  CI.LOWER BOUND: 0.0009812456  CI.UPPER BOUND: 0.002224666  These confidence intervals overlap.  2.  I used the fisher.test function in R to do an exact test on the proportions  fisher.test(rbind(c(9690,5),c(26354,39)))  p-value = 0.01716 --> significant  How can it be that if the confidence intervals overlap so strongly, the p-value is still significant.  Thanks in advance for any help.  Yours sincerely, Martin Rijlaarsdam  @whuber: thanks for pointing me to this answer. I have seen this thread but I was wondering if such a big mismatch between CI and significance of the test is to be expected and if the answer holds for proportions as well. Thanks. Best wishes, Martin
It is well known that confidence intervals and testing statistical hypothesis are strongly related. My questions is focused on comparison of means for two groups based on a numerical variable. Let's assume that such hypothesis is tested using t-test. On the other side, one can compute confidence intervals for means of both groups. Is there any relation between overlapping of confidence intervals and the rejection of the null hypothesis that means are equal (in favor of the alternative that means differ - two-sided test)? For example, a test could reject the null hypothesis iff the confidence intervals do not overlap.
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