Suppose women may have Blue or Red eyes. We make a first cheap research on a 100 women checking just one eye of each woman. So after the research we have results:
At least one eye Blue have avg height 177 cm (50 women)
At least one eye Red have avg height 164cm (70 women)
Since we noticed that one women may have eyes of different color we decided to make second more accurate research. This time we checked both eyes of each woman. Women with eyes:
Blue Blue have avg height 180 cm (30 women)
Red Red have avg height 160 cm (50 women)
Blue Red (sequence does not matter) have avg height 173 cm (20 women)
Now the question. After making the first research we might expect that women with different colors of eyes will have avg height of (177+164)/2=170 cm. It turned out in the second research that they had 3 cm more. What kind of statistical test can determine that 170=173 if we have 20 women of that kind? I would like to know if the BlueRed combination is something totally different feature or it is determined by Blue and Red.
In the same way we may ask - can we expect that women with double Red eyes will have average height (164+164)/2=164cm? It turned out that they had 160cm.
Should we use Chi-squared test? But the problem is that we have a 100 women and 50 Blue and 70Red (don't sum up). Or maybe should we use a two-sample location test of the null hypothesis that the means of two populations are equal? But then, we use the same sample of a 100 women in both cases. How can we check the independence of Blue and Red?