In the chi-square test for independence,expected values and observed values are used,just like in the chi-square goodness-of-fit test.The difference is that in the test of independence,there are three dimensions involved,instead of one.

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The chi-square test for independence involves two dimensions (variables) that are being tested for association, not three. Each dimension corresponds to a different categorical variable.

In the Chi-Square Test for Independence,expected Values and Observed Values