I'm a high school maths teacher and I was helping a pupil who had collected data for her project. I summerise the information in the table below.
$\begin{array}{c|cc|c} & Y & not Y & \\\hline X &360 & 83& 443\\ not X &46 &19 &65\\\hline & 406& 102& 508 \end{array}$
I advised her to do a chi-squared test to test the dependency between the variables X and Y. The table of expected values looks like:
$\begin{array}{c|cc|c} & Y & not Y & \\\hline X &354 & 89& 443\\ not X &52 &13 &65\\\hline & 406& 102& 508 \end{array}$
Comparing these two tables, my first thoughts were that there would be no association. Yet the calculated p-value is 0.048 which seems really rather small and goes against my intuition. It seems the bottom right cell is the one that is causing the effect (I suppose 6 objects from 19 is a fair proportion). I wondered if I have missed an assumption, and searched online and found (wikipedia) that the expected cell count should be no smaller than say 5 or 10, that is absolute values, but there is no mention made of relative count, which seems to me to be the deciding factor here.
Further, I played with the numbers a bit and found that the p-value is very susceptible to a small change, for instance if we remove one object from the bottom right cell the p-value jumps to 0.079.
So my questions:
- Is there something wrong with this analysis?
- Should we conclude that there is an association (assuming 5% significance)?