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I know in studies where relative risk or odds ratio is calculated, if confidence interval overlaps 1, then the P-value is not significant. Is similar relation true for following situations as well?

Consider 2 sets of values with means m1 and m2. I think, if 95% confidence intervals of m1 and m2 are not overlapping, then the p-value for an unpaired t-test comparing them will yield P<0.05, indicating that they are not from same population. Is this true?

Or does one need only to check if m2 lies outside the 95% confidence intervals of m1 (or m1 lies outside 95% confidence intervals of m2), then the p-value of unpaired t-test between 2 sets will yield P<0.05?

Thanks for your insight.

Edit: A corollary of above (as discussed in the comments) is that if notches of 2 boxplots do not overlap, there must be a significant difference between the them.

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Do not use this overlapping criterion because it's not exactly fool-proof. Sometimes the two intervals can overlap and yet the mean difference is statistically significant (p-value is taken from t-test):

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And sometimes one of the means can be excluded from the other group's interval and yet the difference is not statistically significant:

enter image description here

So, there can be weird exceptions. The equivalent statistics you are looking for is the mean difference and its 95% CI. If that interval contains 0, then the p-value will be bigger than 0.05.

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