Question

why does the goodness of fit test and test of independence use the same formula? What's the logic behind it?

Answer 1

The math of the chi-square test of independence is the same as for the chi-square test of goodness-of-fit, only the method of calculating the expected frequencies is different. For the goodness-of-fit test, you use a theoretical relationship to calculate the expected frequencies.

In the test of independence, observational units are collected at random from a population and two categorical variables are observed for each unit..In the goodness-of-fit test there is only one observed variable.

But basically both test have same logic.

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