Download Chi-Square Test for Goodness of Fit and Independence: An Overview and more Slides Statistics in PDF only on Docsity! 1 The Chi-Square Tests for Goodness of Fit And Independence The Chi-Square I. Introduction II. Expected versus Observed Values III. Distribution of X 2 IV. Interpreting SPSS printouts of Chi-Square V. Reporting the Results of Chi-Square VI. Assumptions of Chi-Square Introduction Often when we are testing hypotheses, we only have frequency data. Our hypothesis concern the distributions of the frequencies across various categories. Examples: Are there an equal number of males and females in a group? Are Republicans more likely to be Fundamentalist Christians than Democrats? Docsity.com 2 Introduction With these data we have the number of people of a certain type in a category. This is qualitative, not quantitative date. The scale of measurement is nominal. Compare this to age as a variable. Age is a quantitative variable, measured on a ratio scale. Introduction If one were to ask are Republicans older than Democrats, then one could measure the age of a sample of people in each group, calculate the means of each sample, and test if the difference in the sample means is statistically significant (i.e., the sample means represent a difference in the population mean). Introduction Compare this to the question: “Are Republicans more likely to be males than Democrats?” Our sample would contain a number of males and females. We would not want to calculate a mean gender. Docsity.com 5 Expected versus Observed Values 6040 4258 Males Females Republicans: Observed Expected Χ2 = (58-40)2 + (42-60)2 40 60 Χ2 = 8.1 + 5.4 = 13.5 Distribution of X 2 Large values of X 2 are unlikely to be observed by chance alone (null hypothesis). Distribution of X 2 Shape of the distribution depends on the degrees of freedom. Docsity.com 6 Distribution of X 2 The degrees of freedom are determined by the number of rows and columns in the table. If there is only one row, df = C-1 With more than one row, df = (R-1)(C-1) R = number of rows. C = number of columns. In our example, df = 1 Distribution of X 2 With two dimensions: 2 X 2 Chi-Square Gender and Party Affiliation (observed values) Males Females Republicans Democrats 8070 4258 Totals 100 150 Totals 128 122 250 Null hypothesis: counts will be equally distributed Across the cells. Docsity.com 7 With two dimensions: 2 X 2 Chi-Square Gender and Party Affiliation (expected values) Males Females Republicans Democrats 150*122/250 = 73.2 150*128/250 = 76.8 100*122/250 = 48.8 100*128/250 = 51.2 Totals 100 150 Totals 128 122 250 Use these values to calculate Chi Square: Χ2 = ∑ (fo -fe)2 fe Interpreting SPSS printouts of Chi-Square Data Structure: Case Processing Summary Cases Valid Missing Total N Percent N Percent N Percent Party * Gender 250 100.0% 0 .0% 250 10 Party * Gender Crosstabulation Gender male female Total Party RepublicanCount 58 42 100 Expected Count 51.2 48.8 100.0 Democrat Count 70 80 150 Expected Count 76.8 73.2 150.0 Total Count 128 122 250 Expected Count 128.0 122 Interpreting SPSS printouts of Chi-Square Docsity.com
