# How to interpret the overlap or non-overlap of two related confidence intervals

Say I want to determine if there is a relation/association between birth weights in male and female populations.

Hence I calculate the mean for birth weights in males and females and a calculate 95% CI for each using this formula:

CI = mean +/-  1.96 x SE


I then see if the female group birth weight mean lies within the male CI and vice versa.

If, in the typical scenario, both genders lie within their opposite genders CI then gender is associated/related with birth weight.

However, if not, then gender and birth weight are not associated but are significantly different.

My question is what happens if, e.g., the female population lies in the male CI, but the male population does not lie in the female CI, is it possible?

Is it due to a statistical error?

• Please observe that (caeteris paribus) (1) the widths of confidence intervals tend to shrink with increasing sample size and can be made arbitrarily small and (2) your question presents no constraints on sample size at all. Thus, for instance, if group A were large, it could easily happen that the group B mean would not lie within the CI for A because that CI could be very narrow compared to the sampling variation of the mean of B. That answers your first question. For the second: what do you mean by "associated"?
– whuber
Jan 18, 2013 at 22:49
• hi as im doing medical statistics (beginner level) say i want to determine if there is an association between birthweights in males and female populations. Hence i calculate the mean for male and females and calculate 95% CI for each using this formula: CI = mean +/- 1.96 x SE i then see if the female group mean lies withing the male CI and vice versa. If in the typocal scenario, both genders lie within their opposite genders ci then: If they both lie with the CI of the r Jan 18, 2013 at 23:20
• i rephrased the question, and added a specific example (same as my comment up here) but it didnt let me write the entire thing...so as i said i rephrased question using an example Jan 18, 2013 at 23:31

You make a null hypothesis, $H_0$, which in this case would be the mean birth weight of males and females is the same, and you calculate the probability, if $H_0$ were true, of the means of your two samples not being closer than they actually are. That is the mythical p-value, which, if small enough, allows you to more or less confidently reject the null hypothesis (and get your paper published).
Notice that you can never prove $H_0$, only fail to disprove it, which is not the same. In your case, you cannot say that males and females have the same mean birth weight, only that there is not enough evidence to say they are different...