Statistical Analyses: Chi-square test Psych 250 Winter 2013

Types of Measures / Variables Nominal / categorical –Gender, major, blood type, eye color Ordinal –Rank-order of favorite films; Likert scales? Interval / scale –Time, money, age, GPA

Variable TypeExampleCommonly-used Statistical Method Nominal by Nominalblood type by genderChi-square Scale by NominalGPA by gender GPA by major t-test Analysis of Variance Scale by Scaleweight by height GPA by SAT Regression Correlation Main Analysis Techniques

Question Do men and women differ in the % that choose jail time vs. probation only?

Variable TypeExampleCommonly-used Statistical Method Nominal by Nominal (categorical by categorical) blood type by gender Chi-square Scale by NominalGPA by gender GPA by major t-test Analysis of Variance Scale by Scaleweight by height GPA by SAT Regression Correlation Main Analysis Techniques

Stat Analysis / Hypothesis Testing 1.Form of the relationship 2.Statistical significance

Variables: Categorical by Categorical Form of the relationship:Cross-tab = two-way table Statistical Significance:Chi Square [ if n very small  Fisher’s exact test ]

Example: cross-tab Probation n = 24 Jail n = 16 Males n = Females n =

Example: cross-tab Probation n = 24 Jail n = 16 Males n = % 4 20% Females n = % 12 60%

Example Men more likely to choose probation in the sample Can we infer men in general more likely to choose probation?  Statistical Significance

Statistical Significance Q: Is this a “statistically significant” difference? Can the “null hypothesis” be rejected? Null hypothesis: there are NO differences between men and women in sentencing

Universe n = ∞ M: ?% probation F: ?% probation Sample n = 40 M: 80% probation F: 40% probation sample inference

Universe n = ∞ Null Hypothesis: M% = F% Sample n = 40 M: 80% probation F: 40% probation sample inference

Logic of Statistical Inference 1. If the Null Hypothesis is True… … what are the expected frequencies for Men and Women in any sample? 2. Do the frequencies in my sample (n = 40) differ from the expected frequencies?

Testing Null Hypothesis: Expected Frequencies Probation n = 24 Jail n = 16 Males n = 20 exp: 12exp: 8 Females n = 20 exp: 12exp: 8

Probation n = 24 Jail n = 16 Males n = exp: 12 4 exp: 8 Females n = 20 8 exp: exp: 8 Observed & Expected Frequencies

Logic of Statistical Inference What is the probability of drawing the observed sample (M = 16 probation vs. F = 8 probation) from a universe with no differences? If probability very low, then differences in sample likely reflect differences in universe Then null hypothesis can be rejected; difference in sample is statistically significant

Statistical Significance If probability of obtaining my sample is less than 5 in 100, the null hypothesis can be rejected, and it can be concluded that the difference also exists in the universe. p <.05 The finding from the sample is statistically significant

Strategy Draw an infinite number of samples of n = 40, and graph the distribution of their male vs. female probation %-s

Null Hyp: M = 60% probation F = 60% probation M: 60% F: 50% Samples of n = 40 Universe n = ∞ M: 80% F: 40% M: 70% F: 70% M: 50% F: 65%

Chi Square Distribution 2.5% of area M % > F % 2.5% of area F % > M % M % = F %

Statistical Significance If probability of obtaining my sample is less than 5 in 100, the null hypothesis can be rejected, and it can be concluded that the difference also exists in the universe. p <.05 The finding from the sample is statistically significant

Testing Null Hypothesis: Sample with small difference Probation n = 24 Jail n = 16 Males n = exp: 12 7 exp: 8 Females n = exp: 12 9 exp: 8

Universe N = ∞ M = 60% probation F = 60% probation Sample N = 40 M = 65% F = 55% sample p <.05 ?

Chi Square p =.519 Probation n = 24 Jail n = 16 Males n = exp: 12 7 exp: 8 Females n = exp: 12 9 exp: 8

Small Difference p =.519 Over 50% of samples drawn from null hypothesis universe will have differences this large (65% vs. 55%) Difference is not statistically significant

Probation n = 24 Jail n = 16 Males n = exp: 12 4 exp: 8 Females n = 20 8 exp: exp: 8 Testing Null Hypothesis: Sample with large differences

Universe N = ∞ M = 60% probation F = 60% probation Sample N = 40 M = 80% F = 40% sample p <.05 ?

Chi Square p =.010 Probation n = 24 Jail n = 16 Males n = exp: 12 4 exp: 8 Females n = 20 8 exp: exp: 8

Report Findings “Men were found to choose probation more frequently than women: 80% of the time vs. 40% of the time (df = 1, χ 2 = 6.67, p. <,05).” “Men chose probation 80% of the time, and women only 40% of the time, a difference which was statistically significant (df = 1, χ 2 = 6.67, p. <,05).”