Statistical Analyses: Chi-square test

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. Main Analysis Techniques

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

5. Main Analysis Techniques

6. Stat Analysis / Hypothesis Testing • Form of the relationship • 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

9. Example: cross-tab

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 sample inference M: 80% probation F: 40% probation

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

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

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

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

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

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

25. Chi Square p = .519

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. 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

30. Report Findings • “Men were found to choose probation more frequently than women: 80% of the time vs. 40% of the time (df = 1, χ2 = 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, χ2 = 6.67, p. < ,05).”