INDEPENDENCE AND DEPENDENCE (ASSOCIATION)Conditional percentages refer to a sample data distribution, conditional on a category. They form the conditional distribution. If the probabilities for two different categorical variables are the same in the same category, then these variables are independent. If the probabilities for two different categorical variables differ, then these variables are dependent. Dependence refers to the population, so if there is barely any difference between two categorical variables in a sample, it could be independent, even though they differ.TESTING CATEGORICAL VARIABLES FOR INDEPENDENCEThe expected cell count is the mean of the distribution for the count in any particular cell. The formula for the expected cell count is the following:The chi-squared statistic summarizes how far the observed cell counts in a contingency table fall from the expected cell counts for a null hypothesis. It is the test statistic for the test of independence. The formula for the chi-squared statistic is: The sampling distribution using the chi-squared statistic is called the chi-squared probability distribution. The chi-squared probability distribution has several properties:Always positiveShape depends on degrees of freedomMean equals degrees of freedomAs degrees of freedom increases the distribution becomes more bell shapedLarge chi-square is evidence against independenceThe degrees of freedom in a table with r rows and c columns can be calculated as following:If a response variable is identified and the population conditional distributions are identical, they are said to be homogeneous. The chi-squared test is then referred to as a test of homogeneity. The degrees of freedom value in a chi-squared test indicates how many parameters are needed to determine all the comparisons for describing the contingency table. The chi-squared test can test for independence, but it cannot provide information about the strength and the...