Lecture+14+SU10+_Chi+square+and+ANOVA_

Lecture+14+SU10+_Chi+square+and+ANOVA_ - Chi Square Test...

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02/23/11 Soc 210 summer 2010 1 Chi Square Test and Analysis of Variance (ANOVA) Sociology 210
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02/23/11 Soc 210 summer 2010 2 Example of Independence Population Cross-Classification with Statistical Independence (percentages are the same in each column) Party Identification Ethnic Group Democrat Independent Republican Total White 440 (44%) 140 (14%) 420 (42%) 1000 (100%) Black 44 (44%) 14 (14%) 42 (42%) 100 (100%) Hispanic 110 (44%) 35 (14%) 105 (42%) 250 (100%) Total 594 (44%) 189 (14%) 567 (42%)
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02/23/11 Soc 210 summer 2010 3 Chi Square Test of Independence The definition of statistical independence refers to the population – two variables are independent if they are independent in the population Note that even if the two populations are independent, we would not necessarily expect no relationship in any one sample Because of sampling variability (i.e., sampling error), each sample percentage differs from the true population percentage The question is whether our observed sample differences reflect true population differences, or whether they are due to sampling variability To answer this, we conduct the following hypothesis test H 0 : The variables are statistically independent in the population H a : The variables are statistically dependent in the population This is a test for the association between two categorical variables
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02/23/11 Soc 210 summer 2010 4 Expected Frequencies for Independence The chi-square test compares the observed frequencies in the cells of a contingency table with the frequencies that would be expected under the null hypothesis of independence between the two variables – Let f o denote an observed frequency in a cell of a table – Let f e denote an expected frequency – the count expected in a cell if the two variables were independent (given the row/column totals) Calculation of an expected frequency – The expected frequency, f e , of a cell equals the product of the row marginal totals ( f r ) and column marginal totals ( f c ) for that cell, divided by the total sample size c c r r c e f f n f n f f f ) percent row ( = = = r r c r c e f f n f n f f f ) percent col ( = = =
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02/23/11 Soc 210 summer 2010 5 Example: The expected frequency for Male and Republican, f e = (416)(403)/980 = 171.1 Try to understand the intuition behind expected frequencies: Notice that 416 out of 980 people in the sample (42.4%) identify as Republicans. If the variables were independent, we would expect that 42.4% of males and 42.4% of females identify as Republicans Thus, if the two variables were independent, 42.4% of the 403 males would identify as Republicans, and the expected frequency for the cell would be f e = Party Identification by Gender, with Expected Frequencies in Parentheses Party Identification Gender Democrat Independent Republican Total Female 279 (261.4) 73 (70.7) 225 (244.9) 577 Male 165 (182.6) 47 (49.3) 191 (171.1) 403 Total 444 120 416 980
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02/23/11 Soc 210 summer 2010 6
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This note was uploaded on 02/21/2011 for the course SOCIOLOGY 210 taught by Professor Ybarra during the Summer '10 term at University of Michigan.

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Lecture+14+SU10+_Chi+square+and+ANOVA_ - Chi Square Test...

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