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Most articles that I have found on the subject of testing for the independence of two events state if $P (A \cap B) = P (A) P(B)$, then $A$ and $B$ are independent. If I only had estimates of $P (A\cap B)$, $P (A)$ and $P(B)$ collected from sample data rather exact probabilities calculated using the entire population, will this test for independence still work? If for example, if $P (A\cap B)$ was almost equal to $P (A) \times P(B)$, can I assume independence? Should I be using the chi-square test for independence instead?

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  • $\begingroup$ What does your sample data consist of? $\endgroup$
    – Glen_b
    Jan 27, 2017 at 3:41
  • $\begingroup$ Not to nit-pick, but what would be defined as "almost equal"? $\endgroup$
    – Bensstats
    Jan 27, 2017 at 4:20
  • $\begingroup$ Ploni Almni, by "almost equal" I mean something akin to the statement "I believe with a confidence of 95% that event A is not dependent on B" $\endgroup$
    – eglinker
    Jan 27, 2017 at 4:33
  • $\begingroup$ Glen_b, I don't have a particular data set to provide for you off hand. What I want to be able to do is make observations of events A and B so that I can get estimates of the probabilities, and then use a hypothesis test infer whether or not events A and B ar independent of each other based on whether or not the difference between P(A&B) and P(A)P(B) is statistically significant. $\endgroup$
    – eglinker
    Jan 27, 2017 at 4:51

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You seem to have observations which each can be classified as either A or not A, and either B or not B. That is exactly the pattern behind a $2\times 2$ contingency table, so just make your contingency table and use a chisquare test.

This is all we can say without more context and details.

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