I have $431$ observations of nursing care and am interested to know if touching a patient influenced their handwashing compliance. I have a column for Handwash (Yes/No) and one for Patient contact (Yes/No).
I calculated the probability of Handwash (H) given Patient contact (C) $\Rightarrow \mathbb{P}(H\cap C) = \dfrac{115}{431}$. Fine, now how do I treat that? So how can I put some statistical value or inference to this probability? (Any suggestions or ideas at your discretion)
Your hypothesis is that handwashing is like a coin flip-it is independently done or not with $50\%$ probability. The variance of a coin flip is $\frac 14$, so the variance of $431$ coin flips is $\frac {431}4$ and (using the normal approximation) the standard deviation is the square root of this, about $10.4$. You are about $10$ standard deviations away.