2 - ECMT 1020 Summer School 2009 Chi-Square Tests...

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Chi-Square Tests ECMT 1020 Summer School 2009
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Contingency Tables Contingency Tables Useful in situations involving multiple population proportions Used to classify sample observations according to two or more characteristics Also called a cross-classification table.
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Contingency Table Example Left-Handed vs. Gender Dominant Hand: Left vs. Right Gender: Male vs. Female 2 categories for each variable, so called a 2 x 2 table Suppose we examine a sample of size 300
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Contingency Table Example Sample results organized in a contingency table: (continued) Gender Hand Preference Left Right Female 12 108 120 Male 24 156 180 36 264 300 120 Females, 12 were left handed 180 Males, 24 were left handed sample size = n = 300:
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χ 2 Test for the Difference Between Two Proportions If H 0 is true, then the proportion of left-handed females should be the same as the proportion of left-handed males The two proportions above should be the same as the proportion of left-handed people overall H 0 : π 1 = π 2 (Proportion of females who are left handed is equal to the proportion of males who are left handed) H 1 : π 1 ≠ π 2 (The two proportions are not the same – Hand preference is not independent of gender)
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The Chi-Square Test Statistic where: f o = observed frequency in a particular cell f e = expected frequency in a particular cell if H 0 is true χ 2 for the 2 x 2 case has 1 degree of freedom (Assumed: each cell in the contingency table has expected frequency of at least 5) - = χ cells all e 2 e o 2 f ) f f ( The Chi-square test statistic is:
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Decision Rule χ 2 χ 2 U Decision Rule: If χ 2 > χ 2 U , reject H 0 , otherwise, do not reject H 0 The χ 2 test statistic approximately follows a chi- squared distribution with one degree of freedom 0 α Reject H 0 Do not reject H 0
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Computing the Average Proportion Here: 120 Females, 12 were left handed 180 Males, 24 were left handed i.e., the proportion of left handers overall is 0.12, that is, 12% n X n n X X p 2 1 2 1 = + + = 12 . 0 300 36 180 120 24 12 p = = + + = The average proportion is:
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Finding Expected Frequencies To obtain the expected frequency for left handed females, multiply the average proportion left handed (p) by the total number of females To obtain the expected frequency for left handed males, multiply the average proportion left handed (p) by the total number of males If the two proportions are equal, then P(Left Handed | Female) = P(Left Handed | Male) = .12 i.e., we would expect (.12)(120) = 14.4 females to be left handed (.12)(180) = 21.6 males to be left handed
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Observed vs. Expected Frequencies Gender Hand Preference Left Right Female Observed = 12 Expected = 14.4 Observed = 108 Expected = 105.6 120 Male Observed = 24 Expected = 21.6 Observed = 156 Expected = 158.4 180 36 264 300
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Gender Hand Preference Left Right Female Observed = 12 Expected = 14.4
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2 - ECMT 1020 Summer School 2009 Chi-Square Tests...

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