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Statistical Analyses t-tests

Statistical Analyses t-tests. Psych 250 Winter, 2013. Hypothesis: People will give longer sentences when the victim is female. Independent Variable: Gender of the Victim Dependent Variable: Length of Sentence. Types of Measures / Variables. Nominal / categorical

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Statistical Analyses t-tests

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  1. Statistical Analysest-tests Psych 250 Winter, 2013

  2. Hypothesis: People will give longer sentences when the victim is female.

  3. Independent Variable:Gender of the VictimDependent Variable:Length of Sentence

  4. 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

  5. Main Analysis Techniques

  6. Main Analysis Techniques

  7. Stat Analysis / Hypothesis Testing • Form of the relationship • Statistical significance

  8. Variables:Scale by Categorical • Form of the relationship: Means of each category (M & F victim) • Statistical Significance: Independent samples t-test

  9. Means observed in Sample Victim Gender Average Sentence Male 6 months Female 16 months

  10. Statistical Signficance • Q: Is this a “statistically significant” difference? • Can the “null hypothesis” be rejected? Null hypothesis: there are NO differences in sentencing for male vs. female victims

  11. Universe n = ∞ Sample n = 40 sample inference M victim: 6 months F victim: 16 months

  12. Logic of Statistical Inference • What is the probability of drawing the observed sample (M = 6 months vs. F = 16 months) 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

  13. Strategy • Draw an infinite number of samples of n = 40, and graph the distribution of their male victim / female victim differences

  14. Samples of n = 40 Universe n = ∞ M: 13 F: 9 Null Hyp: M = 11 months F = 11 months M: 6 F: 16 M: 11 F: 11 M: 8 F: 14

  15. T-testSampling distribution: Mean difference Function of: 1) difference in means 2) variance (dispersion around mean)

  16. Possible Sample -- 1 Male VictimFemale Victim 1 2 3 4 5 6 . . . 16

  17. Possible Sample -- 2 Male Victim Female Victim 1 2 3 4 5 6 . . . 16

  18. Frequency Distribution Mean = 11

  19. Variance  x i - Mean )2 Variance = s2 = ----------------------- N  x i - Mean )2 but:s2 = ----------------------- N - 1 Standard Deviation = s =  variance

  20. Calculating Variance Mean = 11

  21. Variance

  22. t distribution • Sampling distribution of a difference in means • Function of mean difference & “pooled” variance (of both samples) mean1 – mean2 t = -------------------------------- sp√ (1/n1) + (1/n2)

  23. Samples of n = 40 Universe n = ∞ mean dif & var Null Hyp: M = 11 months F = 11 months mean dif & var mean dif & var mean dif & var

  24. Samples of n = 40 Universe n = ∞ t Null Hyp: M = 11 months F = 11 months t t t

  25. t distribution 2.5% of area 2.5% of area

  26. Statistical Significance • If probability 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

  27. SPSS t-test Output 1. Read means 3. Read p value 2. Read Levene’s Test

  28. Report Findings • “Assailants were given an average sentence of 16 months when the victims were female, compared to 6 months when the victims were male (df = 46, t = 3.13, p. < .005).” • “Respondents gave longer sentences when the victims were female (16 months) than when they were male (6 months), a difference that was statistically signficant (df = 46, t = 3.13, p. < .005).”

  29. Statistical Analysesanalysis of variance( ANOVA ) Psych 250 Winter, 2011

  30. Analysis of Variance

  31. Dep Var: Length of SentenceIndep var: Major Mean = 14.6 Variance = 212.4

  32. Form of Relationship(differences seen in sample)

  33. Length of Sentence by Major • Nat sci 14.3 • Soc sci 7.4 • Art & Hum 11.0

  34. Statistical Inference( generalize from sample to universe? )

  35. Universe n = ∞ Sample n = 40 sample inference Nat sci = 14.3 Soc sci = 7.4 A & H = 11.0

  36. Possible Sample -- 1 Social ScienceArt & HumanNatural Science 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

  37. Possible Sample -- 2 Social ScienceArt & HumanNatural Science 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

  38. ANOVA Logic • Calculate ratio of “between-groups” variance to “within-groups” variance • Estimate the sampling distribution of that ratio: F distribution • If the probability that the ratio in sample could come from universe with no differences in group means is < .05, can reject null hypothesis and infer that mean differences exist in universe

  39. ANOVA Logic • Between groups: nsocsci(Meansocsci - Mean)2 + narthum(Meanarthum - Mean)2 +nnatsci(Meannatsci – Mean)2 / df • Within groups: (ni – Meansocsci)2 + (ni - Meanarthum)2 + (ni - Meannatsci)2 / df

  40. F ratio between groups mean squares F = within groups mean squares

  41. Samples of n = 40 Universe n = ∞ f Null Hyp: Nat sci = 11 months Soc sci = 11 months Art-Hum = 11 months f f f

  42. f Distributions

  43. ANOVA: sentence by major

  44. ANOVA: sentence by majorsimulated data

  45. ANOVA: sentence by majorsimulated data

  46. Write Findings “Social science majors assigned sentences averaging 7.4 years, arts and humanities students 10.3 years, and natural science students 14.3 years, but these differences were not statistically significant (df = 2, 42, F = 1.35, p < .30).”

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