#### Chi-square test for independence

Test for Independence:

In test for independence, the claim is that the column and row variables are independent of one other. This is the null hypothesis.

The multiplication rule states that when two events were independent then the probability of occurring both was the product of probabilities of each occurring. This is the key to working the test for independence. When you end up refusing the null hypothesis then the supposition should have been wrong and the column and row variable are dependent. Keep in mind that, all hypothesis testing is completed beneath the supposition that null hypothesis is true.

The test statistic employed is similar as the chi-square goodness-of-fit test. The principle behind the test for independence is similar as the principle behind the goodness-of-fit test. The test for independence is for all time a right tail test.

However, you can think of the test for independence as the goodness-of-fit test where the data is arranged into tabular form. This table is termed as contingency table.

The test statistic consists of a chi-square distribution whenever the given assumptions are met:

•    The data obtained are from random sample.
•    The expected frequency of each and every category should be at least 5.

The given are the properties of the test for independence:

a) Data are the observed frequencies.

b) Data is arranged to a contingency table.

c) The degree of freedom is the degree of freedom for row variable times the degrees of freedom for the column variable. This is not one less than the sample size; it is the product of two degrees of freedom.

d) This is always a right tail test.

e) This consists of a chi-square distribution.

f) The predicted value is calculated by taking the row total times the column total and dividing by grand total.

g) The value of test statistic does not change when the order of columns or rows are switched.

h) The value of test statistic does not change when the columns and rows are interchanged (that is, transpose of matrix).

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