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What is Chi-Square? Used to examine differences in the distributions of nominal data A mathematical comparison between expected frequencies and observed frequencies Theoretical, or Expected, Frequencies : developed on the basis of some hypothesis Observed Frequencies : obtained empirically through direct observation

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Assumptions for Chi-Square The samples must have been randomly selected. The data must be in nominal form. The groups for each variable must be completely independent of each other; thus, all cell entries are independent of each other.

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Chi-Square with a Single Variable χ 2 Goodness-of-Fit Test : the fit is said to be good when the observed frequencies are within random fluctuation of the expected frequencies and the computed χ 2 statistic is small and insignificant

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One-Sample Hypotheses Null Hypothesis : There is no significant difference between the observed and expected frequencies. Alternative Hypothesis : There is a significant difference between the observed and expected frequencies.

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Chi-Square with Multiple Variables χ 2 Test of Homogeneity : a test to determine if the frequencies of one variable differ as a function of another variable The independent variable(s) in the χ 2 Test of Homogeneity are called the antecedent variable (s); they are the ones which logically precede the others. The Chi-Square Test can accommodate multiple variables, e.g. 2 x 2 3 X 5 2 x 3 x 5

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Two-Sample Hypotheses Null Hypothesis: The frequency distribution of variable Y does not differ as a result of group membership in variable X. Non-Directional Alternative Hypothesis: The frequency distribution of variable Y does differ as a result of group membership in variable X.

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The Chi-Square Distribution There is a family of χ 2 distributions, each determined by a single degree of freedom value. For a single variable: df = k – 1 For multiple variables: df = (r – 1)(c – 1) Where r = the number of rows c = the number of columns As the degrees of freedom increase, the sampling distribution approaches the normal distribution.

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Computing Chi-Square with a Single Variable To enter the data Create columns for each variable Each variable will have value labels The level of measurement for all variables will be nominal Analyze Nonparametric Chi-Square Move the variable(s) of interest to the Test Variable List Click OK

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Output for a Single Variable

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Computing Chi-Square with More Than One Variable Analyze Descriptive Statistics Crosstabs Move the antecedent (independent) variable(s) to the Row(s) box Move the dependent variable(s) to the Column(s) box Click Statistics Check Chi-Square Click Continue Click OK

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Output for a 2 X 2 Chi-Square

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