1. Statistical Analyses: Chi-square test Psych 250 Winter 2013

2. 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

3. 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

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

5. 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

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

7. 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 ]

8. Example: cross-tab Probation n = 24 Jail n = 16 Males n = 20 164 Females n = 20 812

9. Example: cross-tab Probation n = 24 Jail n = 16 Males n = 20 16 80% 4 20% Females n = 20 8 40% 12 60%

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

11. 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

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

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

14. 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?

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

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

17. 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

18. 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

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

20. 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%

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

22. 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

23. Testing Null Hypothesis: Sample with small difference Probation n = 24 Jail n = 16 Males n = 20 13 exp: 12 7 exp: 8 Females n = 20 11 exp: 12 9 exp: 8

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

25. Chi Square p =.519 Probation n = 24 Jail n = 16 Males n = 20 13 exp: 12 7 exp: 8 Females n = 20 11 exp: 12 9 exp: 8

26. 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

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

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

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

30. 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).”